3.4.97 \(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [397]

Optimal. Leaf size=354 \[ -\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {16 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/5*(c*x^2+b*x)^(3/2)/e/(e*x+d)^(5/2)+2/5*(b^2*e^2-16*b*c*d*e+16*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/e^4/(-b*e+c*d)/(1+e*x/d)^(1/2)/(
c*x^2+b*x)^(1/2)-16/5*(-b*e+2*c*d)*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)/e^4/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/5*(c*d^2*(-7*b*e+8*c*d)+e*(b^2*e^2
-10*b*c*d*e+10*c^2*d^2)*x)*(c*x^2+b*x)^(1/2)/d/e^3/(-b*e+c*d)/(e*x+d)^(3/2)

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Rubi [A]
time = 0.49, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {746, 824, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 d e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}-\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+c d^2 (8 c d-7 b e)\right )}{5 d e^3 (d+e x)^{3/2} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

(-2*(c*d^2*(8*c*d - 7*b*e) + e*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(5*d*e^3*(c*d - b*e)*
(d + e*x)^(3/2)) - (2*(b*x + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) + (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[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)
])/(5*d*e^4*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (16*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt
[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(5*e^4*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 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)^{7/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx}{5 e}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {2 \int \frac {-\frac {1}{2} b c d (8 c d-7 b e)-\frac {1}{2} c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{5 d e^3 (c d-b e)}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{5 e^4}+\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{5 d e^4 (c d-b e)}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {\left (8 c (2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{5 e^4 \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 {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{5 d e^4 (c d-b e) \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\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 {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (8 c (2 c d-b e) \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}{5 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\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 {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {16 \sqrt {-b} \sqrt {c} (2 c d-b e) \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 )}{5 e^4 \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 = 10.87, size = 369, normalized size = 1.04 \begin {gather*} -\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (b^2 e^4 x^2-b c d e \left (7 d^2+16 d e x+11 e^2 x^2\right )+c^2 d^2 \left (8 d^2+18 d e x+11 e^2 x^2\right )\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) (b+c x) (d+e x)+i b e \left (16 c^2 d^2-16 b c d e+b^2 e^2\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^2 d^2-9 b c d e+b^2 e^2\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 )}{5 b d e^4 (c d-b e) x^2 (b+c x)^2 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(b^2*e^4*x^2 - b*c*d*e*(7*d^2 + 16*d*e*x + 11*e^2*x^2) + c^2*d^2*(8*d
^2 + 18*d*e*x + 11*e^2*x^2)) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x
)*(d + e*x) + I*b*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*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^2*d^2 - 9*b*c*d*e + b^2*e^2)*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)])))/(5*b*d*e^4*(c*d - b*e)*x^2*(b + c*x
)^2*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1884\) vs. \(2(300)=600\).
time = 0.47, size = 1885, normalized size = 5.32

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 e^{6} \left (x +\frac {d}{e}\right )^{3}}-\frac {4 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 d \left (b e -c d \right ) e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c \left (2 b e -3 c d \right )}{e^{4}}-\frac {2 c \left (b e -2 c d \right )}{5 e^{4}}+\frac {b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}}{5 e^{4} d}-\frac {b \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 e^{3} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (\frac {c^{2}}{e^{3}}-\frac {c \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 e^{3} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(646\)
default	\(\text {Expression too large to display}\)	\(1885\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/5*(x*(c*x+b))^(1/2)*(-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^2*c^2*d^3*e^2*x+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*c^3*d^4*e*x-34*((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*d^2*e^3*x+64*((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^2*d^3*e^
2*x-32*((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^3*d^4*e*x+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*d*e^4*x+32*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e+8*((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*d^3*e^2-24*((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^2*d^4*e-17*((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*d^3*e^2+18*c^4*d^3*e^2*x^3+8*c^4*d^4*e*x^2+32*((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^2*d^2*e^3*x^2-16*((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^3*d^3*e^2*x^2+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^3*c*d^2
*e^3*x+8*((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*d*e^4*x^2-24*((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^2*d^2*e^3*x^2+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^3*d^3*e^2*x^2-17*((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*d*e^4*x^2+b^2*c^2*e^
5*x^4+11*c^4*d^2*e^3*x^4+b^3*c*e^5*x^3-11*b*c^3*d*e^4*x^4-11*b^2*c^2*d*e^4*x^3-5*b*c^3*d^2*e^3*x^3-16*b^2*c^2*
d^2*e^3*x^2+11*b*c^3*d^3*e^2*x^2-7*b^2*c^2*d^3*e^2*x+8*b*c^3*d^4*e*x+((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*e^5*x^2+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^3*d^5+((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*d^2
*e^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^3*d^5)/(c*x+b)/x/(b*e-c*d)/c/(e*x+d)^(5/2)/d/e^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)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(x*e + d)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.50, size = 781, normalized size = 2.21 \begin {gather*} -\frac {2 \, {\left ({\left (16 \, c^{3} d^{6} + b^{3} x^{3} e^{6} + 3 \, {\left (2 \, b^{2} c d x^{3} + b^{3} d x^{2}\right )} e^{5} - 3 \, {\left (8 \, b c^{2} d^{2} x^{3} - 6 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{4} + {\left (16 \, c^{3} d^{3} x^{3} - 72 \, b c^{2} d^{3} x^{2} + 18 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} e^{3} + 6 \, {\left (8 \, c^{3} d^{4} x^{2} - 12 \, b c^{2} d^{4} x + b^{2} c d^{4}\right )} e^{2} + 24 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\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 ) + 3 \, {\left (16 \, c^{3} d^{5} e + b^{2} c x^{3} e^{6} - {\left (16 \, b c^{2} d x^{3} - 3 \, b^{2} c d x^{2}\right )} e^{5} + {\left (16 \, c^{3} d^{2} x^{3} - 48 \, b c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x\right )} e^{4} + {\left (48 \, c^{3} d^{3} x^{2} - 48 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} e^{3} + 16 \, {\left (3 \, c^{3} d^{4} x - b c^{2} d^{4}\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^{3} d^{4} e^{2} - 11 \, b c^{2} d x^{2} e^{5} + b^{2} c x^{2} e^{6} + {\left (11 \, c^{3} d^{2} x^{2} - 16 \, b c^{2} d^{2} x\right )} e^{4} + {\left (18 \, c^{3} d^{3} x - 7 \, b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{15 \, {\left (c^{2} d^{5} e^{5} - b c d x^{3} e^{9} + {\left (c^{2} d^{2} x^{3} - 3 \, b c d^{2} x^{2}\right )} e^{8} + 3 \, {\left (c^{2} d^{3} x^{2} - b c d^{3} x\right )} e^{7} + {\left (3 \, c^{2} d^{4} x - b c d^{4}\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)^(7/2),x, algorithm="fricas")

[Out]

-2/15*((16*c^3*d^6 + b^3*x^3*e^6 + 3*(2*b^2*c*d*x^3 + b^3*d*x^2)*e^5 - 3*(8*b*c^2*d^2*x^3 - 6*b^2*c*d^2*x^2 -
b^3*d^2*x)*e^4 + (16*c^3*d^3*x^3 - 72*b*c^2*d^3*x^2 + 18*b^2*c*d^3*x + b^3*d^3)*e^3 + 6*(8*c^3*d^4*x^2 - 12*b*
c^2*d^4*x + b^2*c*d^4)*e^2 + 24*(2*c^3*d^5*x - b*c^2*d^5)*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) + 3*(16*c^3*d^5*e + b^2*c*x^3*e^6 - (16*b*c^2*d*x^3 - 3*b^2*c*d*x^2)*e^5 + (1
6*c^3*d^2*x^3 - 48*b*c^2*d^2*x^2 + 3*b^2*c*d^2*x)*e^4 + (48*c^3*d^3*x^2 - 48*b*c^2*d^3*x + b^2*c*d^3)*e^3 + 16
*(3*c^3*d^4*x - b*c^2*d^4)*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, 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)) + 3*(8*c^3*d^4*e^2 - 11*b*c^2*d*x^2*e^5 + b^2*c*x^2*e^6 + (11*c^3*d^2*x^2 - 16*
b*c^2*d^2*x)*e^4 + (18*c^3*d^3*x - 7*b*c^2*d^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))/(c^2*d^5*e^5 - b*c*d*x^3
*e^9 + (c^2*d^2*x^3 - 3*b*c*d^2*x^2)*e^8 + 3*(c^2*d^3*x^2 - b*c*d^3*x)*e^7 + (3*c^2*d^4*x - b*c*d^4)*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 {7}{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)**(7/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)/(d + e*x)**(7/2), 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)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(x*e + d)^(7/2), 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 )}^{7/2}} \,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)^(7/2),x)

[Out]

int((b*x + c*x^2)^(3/2)/(d + e*x)^(7/2), x)

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