3.4.92 \(\int (d+e x)^{3/2} (b x+c x^2)^{3/2} \, dx\) [392]

Optimal. Leaf size=521 \[ \frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {16 \sqrt {-b} (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

2/231*(c^2*d^2+13*b*c*d*e-6*b^2*e^2+14*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/c^2/e+2/11*e*(c*x^2
+b*x)^(5/2)*(e*x+d)^(1/2)/c-16/1155*(-2*b*e+c*d)*(-b*e+2*c*d)*(b*e+c*d)*(b^2*e^2-b*c*d*e+c^2*d^2)*EllipticE(c^
(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/c^(7/2)/e^4/(1+e*x/
d)^(1/2)/(c*x^2+b*x)^(1/2)+2/1155*d*(-b*e+c*d)*(-8*b^4*e^4+13*b^3*c*d*e^3+3*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*
c^4*d^4)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d)^(1
/2)/c^(7/2)/e^4/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/1155*(8*c^4*d^4-19*b*c^3*d^3*e+6*b^2*c^2*d^2*e^2-19*b^3*c*d*
e^3+8*b^4*e^4-3*c*e*(-b*e+2*c*d)*(8*b^2*e^2-b*c*d*e+c^2*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^3/e^3

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Rubi [A]
time = 0.47, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {756, 828, 857, 729, 113, 111, 118, 117} \begin {gather*} -\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{231 c^2 e}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{1155 c^3 e^3}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 19*b^3*c*d*e^3 + 8*b^4*e^4 - 3*c*e*(2*c*d -
b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 + 13*b*
c*d*e - 6*b^2*e^2 + 14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(b*x + c*x^2
)^(5/2))/(11*c) - (16*Sqrt[-b]*(c*d - 2*b*e)*(2*c*d - b*e)*(c*d + b*e)*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*e^4*S
qrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e
^2 + 13*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x]
)/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*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 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx &=\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (\frac {1}{2} d (11 c d-5 b e)+3 e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 c}\\ &=\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {4 \int \frac {\left (\frac {3}{4} b d e \left (c^2 d^2+13 b c d e-6 b^2 e^2\right )+\frac {3}{4} e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {-\frac {3}{8} b d e \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4\right )-3 e (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac {\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{1155 c^3 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-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}{1155 c^3 e^4 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{1155 c^3 e^4 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-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}{1155 c^3 e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\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}{1155 c^3 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {16 \sqrt {-b} (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-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 )}{1155 c^{7/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\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 )}{1155 c^{7/2} 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 = 20.03, size = 559, normalized size = 1.07 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (8 b^4 e^4-b^3 c e^3 (19 d+6 e x)+b^2 c^2 e^2 \left (6 d^2+14 d e x+5 e^2 x^2\right )+b c^3 e \left (-19 d^3+14 d^2 e x+205 d e^2 x^2+140 e^3 x^3\right )+c^4 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} \left (-8 \sqrt {\frac {b}{c}} \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\right ) (b+c x) (d+e x)-8 i b e \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\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^5 d^5-21 b c^4 d^4 e+10 b^2 c^3 d^3 e^2+35 b^3 c^2 d^2 e^3-48 b^4 c d e^4+16 b^5 e^5\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 )}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(8*b^4*e^4 - b^3*c*e^3*(19*d + 6*e*x) + b^2*c^2*e^2*(6*d^2 +
14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19*d^3 + 14*d^2*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x
+ 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*(2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*
c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*(b + c*x)*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*
c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*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^5*d^5 - 21*b*c^4*d^4*e + 10*b^2
*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*E
llipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1358\) vs. \(2(461)=922\).
time = 0.46, size = 1359, normalized size = 2.61

method	result	size
default	\(\text {Expression too large to display}\)	\(1359\)
elliptic	\(\text {Expression too large to display}\)	\(1623\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(10*((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^4*c^3*d^3*e^3+35*((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^4*d^4*e^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^5*d^5*e-56*((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^6*c*d*e^5+56*((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*c^2*d^2*e^4+56*((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^5*d^5*e+b^2*c^5*d^3*e^3*x^2-17*b*c^6*d^4*e^2*x^2+8*b^5*c^2*d*e^5
*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^6*c*d*e^5-21*((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^5*c^2*d^2*e^4-56*((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^3*c^4*d^4*e^2-17*b^4*c^3*d*e^5*x^2+239*b^2*c^5*d^2*e^4*x^3-6
*b*c^6*d^3*e^3*x^3+245*b*c^6*e^6*x^6+b^3*c^4*d^2*e^4*x^2+105*c^7*e^6*x^7+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^7*e^6-c^7*d^3*e^3*x^4+2*b^4*c
^3*e^6*x^3+2*c^7*d^4*e^2*x^3+8*b^5*c^2*e^6*x^2+8*c^7*d^5*e*x^2+245*c^7*d*e^5*x^6+145*b^2*c^5*e^6*x^5+145*c^7*d
^2*e^4*x^5-b^3*c^4*e^6*x^4+590*b*c^6*d*e^5*x^5+364*b^2*c^5*d*e^5*x^4+364*b*c^6*d^2*e^4*x^4-6*b^3*c^4*d*e^5*x^3
-19*b^4*c^3*d^2*e^4*x+6*b^3*c^4*d^3*e^3*x-19*b^2*c^5*d^4*e^2*x+8*b*c^6*d^5*e*x+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^6*d^6-16*((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^6*d^6
)/c^5/e^4/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.71, size = 601, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 33 \, b^{2} c^{4} d^{4} e^{2} + 14 \, b^{3} c^{3} d^{3} e^{3} + 33 \, b^{4} c^{2} d^{2} e^{4} - 48 \, b^{5} c d e^{5} + 16 \, b^{6} e^{6}\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 ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, b^{2} c^{4} d^{3} e^{3} + 2 \, b^{3} c^{3} d^{2} e^{4} - 5 \, b^{4} c^{2} d e^{5} + 2 \, b^{5} c e^{6}\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^{6} d^{4} e^{2} + {\left (105 \, c^{6} x^{4} + 140 \, b c^{5} x^{3} + 5 \, b^{2} c^{4} x^{2} - 6 \, b^{3} c^{3} x + 8 \, b^{4} c^{2}\right )} e^{6} + {\left (140 \, c^{6} d x^{3} + 205 \, b c^{5} d x^{2} + 14 \, b^{2} c^{4} d x - 19 \, b^{3} c^{3} d\right )} e^{5} + {\left (5 \, c^{6} d^{2} x^{2} + 14 \, b c^{5} d^{2} x + 6 \, b^{2} c^{4} d^{2}\right )} e^{4} - {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3465 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*((16*c^6*d^6 - 48*b*c^5*d^5*e + 33*b^2*c^4*d^4*e^2 + 14*b^3*c^3*d^3*e^3 + 33*b^4*c^2*d^2*e^4 - 48*b^5*c
*d*e^5 + 16*b^6*e^6)*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) + 24*(2
*c^6*d^5*e - 5*b*c^5*d^4*e^2 + 2*b^2*c^4*d^3*e^3 + 2*b^3*c^3*d^2*e^4 - 5*b^4*c^2*d*e^5 + 2*b^5*c*e^6)*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^
6*d^4*e^2 + (105*c^6*x^4 + 140*b*c^5*x^3 + 5*b^2*c^4*x^2 - 6*b^3*c^3*x + 8*b^4*c^2)*e^6 + (140*c^6*d*x^3 + 205
*b*c^5*d*x^2 + 14*b^2*c^4*d*x - 19*b^3*c^3*d)*e^5 + (5*c^6*d^2*x^2 + 14*b*c^5*d^2*x + 6*b^2*c^4*d^2)*e^4 - (6*
c^6*d^3*x + 19*b*c^5*d^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))*e^(-5)/c^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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