3.4.98 \(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^{9/2}} \, dx\) [398]

Optimal. Leaf size=476 \[ \frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {4 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d^2 e^4 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d e^4 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/7*(c*x^2+b*x)^(3/2)/e/(e*x+d)^(7/2)-4/35*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*EllipticE(c^(1/2)*x^(1
/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/d^2/e^4/(-b*e+c*d)^2/
(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/35*(-b^2*e^2-16*b*c*d*e+16*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(
b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1/2)/d/e^4/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x
^2+b*x)^(1/2)-2/35*(d*(-2*b^2*e^2-5*b*c*d*e+8*c^2*d^2)+e*(b^2*e^2-14*b*c*d*e+14*c^2*d^2)*x)*(c*x^2+b*x)^(1/2)/
d/e^3/(-b*e+c*d)/(e*x+d)^(5/2)+4/35*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*(c*x^2+b*x)^(1/2)/d^2/e^3/(-b*
e+c*d)^2/(e*x+d)^(1/2)

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Rubi [A]
time = 0.58, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {746, 824, 848, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d e^4 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d^2 e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{35 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{35 d^2 e^3 \sqrt {d+e x} (c d-b e)^2}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(4*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[b*x + c*x^2])/(35*d^2*e^3*(c*d - b*e)^2*Sqrt[d + e*x])
- (2*(d*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + e*(14*c^2*d^2 - 14*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(35
*d*e^3*(c*d - b*e)*(d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(3/2))/(7*e*(d + e*x)^(7/2)) - (4*Sqrt[-b]*Sqrt[c]*(2*c
*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*
Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d^2*e^4*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]
*Sqrt[c]*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d*e^4*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x]*(Sqrt[
1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)])), Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 729

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b
*x + c*x^2]), Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {2 \int \frac {-\frac {1}{2} b \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )-\frac {1}{2} c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{35 d e^3 (c d-b e)}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-\frac {1}{2} c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 d^2 e^3 (c d-b e)^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 d e^4 (c d-b e)}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{35 d^2 e^4 (c d-b e)^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{35 d e^4 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{35 d^2 e^4 (c d-b e)^2 \sqrt {b x+c x^2}}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{35 d^2 e^4 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{35 d e^4 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {4 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d^2 e^4 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d e^4 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.57, size = 479, normalized size = 1.01 \begin {gather*} -\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (5 d^3 (c d-b e)^3-8 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)+d (c d-b e) \left (19 c^2 d^2-19 b c d e+b^2 e^2\right ) (d+e x)^2-2 \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (d+e x)^3\right )+\sqrt {\frac {b}{c}} c (d+e x)^3 \left (2 \sqrt {\frac {b}{c}} \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (b+c x) (d+e x)+2 i b e \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^3 d^3-13 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{35 b d^2 e^4 (c d-b e)^2 x^2 (b+c x)^2 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(5*d^3*(c*d - b*e)^3 - 8*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) +
d*(c*d - b*e)*(19*c^2*d^2 - 19*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 2*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2
+ b^3*e^3)*(d + e*x)^3) + Sqrt[b/c]*c*(d + e*x)^3*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b
^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x
)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^
2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]
/Sqrt[x]], (c*d)/(b*e)])))/(35*b*d^2*e^4*(c*d - b*e)^2*x^2*(b + c*x)^2*(d + e*x)^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3266\) vs. \(2(416)=832\).
time = 0.48, size = 3267, normalized size = 6.86

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{7 e^{7} \left (x +\frac {d}{e}\right )^{4}}-\frac {16 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{35 e^{6} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (b^{2} e^{2}-19 b c d e +19 d^{2} c^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{35 d \left (b e -c d \right ) e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (c e \,x^{2}+b e x \right ) \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right )}{35 d^{2} \left (b e -c d \right )^{2} e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c^{2}}{e^{4}}+\frac {c \left (b^{2} e^{2}-19 b c d e +19 d^{2} c^{2}\right )}{35 e^{4} d \left (b e -c d \right )}+\frac {\frac {4}{35} b^{2} d \,e^{2} c -\frac {24}{35} b \,c^{2} d^{2} e +\frac {16}{35} c^{3} d^{3}+\frac {2}{35} b^{3} e^{3}}{e^{4} \left (b e -c d \right ) d^{2}}-\frac {2 b \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right )}{35 e^{3} d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}-\frac {4 \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{35 e^{3} d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(792\)
default	\(\text {Expression too large to display}\)	\(3267\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/35*(x*(c*x+b))^(1/2)*(6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^
(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d*e^6*x^2+6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d^2*e^5*x+((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^4*e^3+15*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^5*e^2-32*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2
*c^3*d^6*e+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b^4*c*d^4*e^3-28*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^5*e^2+40*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^6*e+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c
*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^7*x^3+16*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^7+2*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^5*d^3*e^4-16*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*b*c^4*d^7-84*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3*e^4*x^2+120*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^4*e^3*x^2-48*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5*e^2*x^2+3*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^4*c*d^3*e^4*x+45*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^4*e^3*x-96*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^5*e^2*x+48*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^6*e*x+6*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^3*e^4*x-84*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^3*c^2*d^4*e^3*x+6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*e^5*x^2+((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^6*x^3+15*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^5*x^3-32*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e^4*x
^3+16*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d)
)^(1/2))*b*c^4*d^4*e^3*x^3+2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^6*x^3-28*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^5*x^3+40*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e^4*x^3-16*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^
4*d^4*e^3*x^3+3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e
/(b*e-c*d))^(1/2))*b^4*c*d^2*e^5*x^2+45*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
F(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3*e^4*x^2-96*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^4*e^3*x^2+48*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5*e^2*x
^2+120*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*b^2*c^3*d^5*e^2*x-48*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^6*e*x+2*b^3*c^2*e^7*x^5+16*c^5*d^3*e^4*x^5+2*b^4*c*e^7*x^4+29*c^5*d^
4*e^3*x^4+26*c^5*d^5*e^2*x^3+8*c^5*d^6*e*x^2+4*b^2*c^3*d*e^6*x^5-24*b*c^4*d^2*e^5*x^5+11*b^3*c^2*d*e^6*x^4-32*
b^2*c^3*d^2*e^5*x^4-18*b*c^4*d^3*e^4*x^4+7*b^4*...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(x*e + d)^(9/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 1199, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{4} d^{8} + 2 \, b^{4} x^{4} e^{8} + {\left (3 \, b^{3} c d x^{4} + 8 \, b^{4} d x^{3}\right )} e^{7} + {\left (13 \, b^{2} c^{2} d^{2} x^{4} + 12 \, b^{3} c d^{2} x^{3} + 12 \, b^{4} d^{2} x^{2}\right )} e^{6} - 2 \, {\left (16 \, b c^{3} d^{3} x^{4} - 26 \, b^{2} c^{2} d^{3} x^{3} - 9 \, b^{3} c d^{3} x^{2} - 4 \, b^{4} d^{3} x\right )} e^{5} + 2 \, {\left (8 \, c^{4} d^{4} x^{4} - 64 \, b c^{3} d^{4} x^{3} + 39 \, b^{2} c^{2} d^{4} x^{2} + 6 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} e^{4} + {\left (64 \, c^{4} d^{5} x^{3} - 192 \, b c^{3} d^{5} x^{2} + 52 \, b^{2} c^{2} d^{5} x + 3 \, b^{3} c d^{5}\right )} e^{3} + {\left (96 \, c^{4} d^{6} x^{2} - 128 \, b c^{3} d^{6} x + 13 \, b^{2} c^{2} d^{6}\right )} e^{2} + 32 \, {\left (2 \, c^{4} d^{7} x - b c^{3} d^{7}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left (8 \, c^{4} d^{7} e + b^{3} c x^{4} e^{8} + 2 \, {\left (b^{2} c^{2} d x^{4} + 2 \, b^{3} c d x^{3}\right )} e^{7} - 2 \, {\left (6 \, b c^{3} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{3} - 3 \, b^{3} c d^{2} x^{2}\right )} e^{6} + 4 \, {\left (2 \, c^{4} d^{3} x^{4} - 12 \, b c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{3} c d^{3} x\right )} e^{5} + {\left (32 \, c^{4} d^{4} x^{3} - 72 \, b c^{3} d^{4} x^{2} + 8 \, b^{2} c^{2} d^{4} x + b^{3} c d^{4}\right )} e^{4} + 2 \, {\left (24 \, c^{4} d^{5} x^{2} - 24 \, b c^{3} d^{5} x + b^{2} c^{2} d^{5}\right )} e^{3} + 4 \, {\left (8 \, c^{4} d^{6} x - 3 \, b c^{3} d^{6}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{4} d^{6} e^{2} + 2 \, b^{3} c x^{3} e^{8} + {\left (4 \, b^{2} c^{2} d x^{3} + 7 \, b^{3} c d x^{2}\right )} e^{7} - 8 \, {\left (3 \, b c^{3} d^{2} x^{3} + b^{2} c^{2} d^{2} x^{2}\right )} e^{6} + 2 \, {\left (8 \, c^{4} d^{3} x^{3} - 17 \, b c^{3} d^{3} x^{2} + 2 \, b^{2} c^{2} d^{3} x\right )} e^{5} + {\left (29 \, c^{4} d^{4} x^{2} - 36 \, b c^{3} d^{4} x + b^{2} c^{2} d^{4}\right )} e^{4} + {\left (26 \, c^{4} d^{5} x - 11 \, b c^{3} d^{5}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{105 \, {\left (c^{3} d^{8} e^{5} + b^{2} c d^{2} x^{4} e^{11} - 2 \, {\left (b c^{2} d^{3} x^{4} - 2 \, b^{2} c d^{3} x^{3}\right )} e^{10} + {\left (c^{3} d^{4} x^{4} - 8 \, b c^{2} d^{4} x^{3} + 6 \, b^{2} c d^{4} x^{2}\right )} e^{9} + 4 \, {\left (c^{3} d^{5} x^{3} - 3 \, b c^{2} d^{5} x^{2} + b^{2} c d^{5} x\right )} e^{8} + {\left (6 \, c^{3} d^{6} x^{2} - 8 \, b c^{2} d^{6} x + b^{2} c d^{6}\right )} e^{7} + 2 \, {\left (2 \, c^{3} d^{7} x - b c^{2} d^{7}\right )} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

2/105*((16*c^4*d^8 + 2*b^4*x^4*e^8 + (3*b^3*c*d*x^4 + 8*b^4*d*x^3)*e^7 + (13*b^2*c^2*d^2*x^4 + 12*b^3*c*d^2*x^
3 + 12*b^4*d^2*x^2)*e^6 - 2*(16*b*c^3*d^3*x^4 - 26*b^2*c^2*d^3*x^3 - 9*b^3*c*d^3*x^2 - 4*b^4*d^3*x)*e^5 + 2*(8
*c^4*d^4*x^4 - 64*b*c^3*d^4*x^3 + 39*b^2*c^2*d^4*x^2 + 6*b^3*c*d^4*x + b^4*d^4)*e^4 + (64*c^4*d^5*x^3 - 192*b*
c^3*d^5*x^2 + 52*b^2*c^2*d^5*x + 3*b^3*c*d^5)*e^3 + (96*c^4*d^6*x^2 - 128*b*c^3*d^6*x + 13*b^2*c^2*d^6)*e^2 +
32*(2*c^4*d^7*x - b*c^3*d^7)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c
^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)
/c) + 6*(8*c^4*d^7*e + b^3*c*x^4*e^8 + 2*(b^2*c^2*d*x^4 + 2*b^3*c*d*x^3)*e^7 - 2*(6*b*c^3*d^2*x^4 - 4*b^2*c^2*
d^2*x^3 - 3*b^3*c*d^2*x^2)*e^6 + 4*(2*c^4*d^3*x^4 - 12*b*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 + b^3*c*d^3*x)*e^5 +
(32*c^4*d^4*x^3 - 72*b*c^3*d^4*x^2 + 8*b^2*c^2*d^4*x + b^3*c*d^4)*e^4 + 2*(24*c^4*d^5*x^2 - 24*b*c^3*d^5*x + b
^2*c^2*d^5)*e^3 + 4*(8*c^4*d^6*x - 3*b*c^3*d^6)*e^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e +
b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*e^(-3)/c^3, weierstrassPInv
erse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^
3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) + 3*(8*c^4*d^6*e^2 + 2*b^3*c*x^3*e^8 + (4*b^2*c^2*d*x^3 +
7*b^3*c*d*x^2)*e^7 - 8*(3*b*c^3*d^2*x^3 + b^2*c^2*d^2*x^2)*e^6 + 2*(8*c^4*d^3*x^3 - 17*b*c^3*d^3*x^2 + 2*b^2*c
^2*d^3*x)*e^5 + (29*c^4*d^4*x^2 - 36*b*c^3*d^4*x + b^2*c^2*d^4)*e^4 + (26*c^4*d^5*x - 11*b*c^3*d^5)*e^3)*sqrt(
c*x^2 + b*x)*sqrt(x*e + d))/(c^3*d^8*e^5 + b^2*c*d^2*x^4*e^11 - 2*(b*c^2*d^3*x^4 - 2*b^2*c*d^3*x^3)*e^10 + (c^
3*d^4*x^4 - 8*b*c^2*d^4*x^3 + 6*b^2*c*d^4*x^2)*e^9 + 4*(c^3*d^5*x^3 - 3*b*c^2*d^5*x^2 + b^2*c*d^5*x)*e^8 + (6*
c^3*d^6*x^2 - 8*b*c^2*d^6*x + b^2*c*d^6)*e^7 + 2*(2*c^3*d^7*x - b*c^2*d^7)*e^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)/(d + e*x)**(9/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(x*e + d)^(9/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x)

[Out]

int((b*x + c*x^2)^(3/2)/(d + e*x)^(9/2), x)

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