3.4.93 \(\int \sqrt {d+e x} (b x+c x^2)^{3/2} \, dx\) [393]

Optimal. Leaf size=457 \[ \frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \sqrt {-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

2/9*(e*x+d)^(3/2)*(c*x^2+b*x)^(3/2)/e-2/21*(-b*e+2*c*d)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/c/e-2/315*(-8*b^4*e^4+
7*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)*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^(5/2)/e^4/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+8/315*d*(-b*
e+c*d)*(-b*e+2*c*d)*(-b^2*e^2-2*b*c*d*e+2*c^2*d^2)*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^(5/2)/e^4/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/315*(8*c^3*d^3-15*
b*c^2*d^2*e+3*b^2*c*d*e^2-4*b^3*e^3-6*c*e*(2*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^2/e
^3

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Rubi [A]
time = 0.38, antiderivative size = 457, 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 = {748, 846, 828, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {8 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}+\frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{21 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e*(c^2*d^2 - b*c*d*e + 2*b^2*e^
2)*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d
+ e*x)^(3/2)*(b*x + c*x^2)^(3/2))/(9*e) - (2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3
*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b
*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(2
*c^2*d^2 - 2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*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 748

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 + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 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 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 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 \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {\int \sqrt {d+e x} (b d+(2 c d-b e) x) \sqrt {b x+c x^2} \, dx}{3 e}\\ &=-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \int \frac {\left (\frac {1}{2} b d (c d+3 b e)+\left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {4 \int \frac {-\frac {1}{4} b d \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )-\frac {1}{4} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^4}-\frac {\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 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}{315 c^2 e^4 \sqrt {b x+c x^2}}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{315 c^2 e^4 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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}{315 c^2 e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 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}{315 c^2 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \sqrt {-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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 )}{315 c^{5/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 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 )}{315 c^{5/2} 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 = 16.01, size = 463, normalized size = 1.01 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (-4 b^3 e^3+3 b^2 c e^2 (d+e x)+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\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 )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(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)*(-4*b^3*e^3 + 3*b^2*c*e^2*(d + e*x) + b*c^2*e*(-15*d^2 + 11*
d*e*x + 50*e^2*x^2) + c^3*(8*d^3 - 6*d^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)) - Sqrt[b/c]*(Sqrt[b/c]*(16*c^4*d^4 -
32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*
b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip
ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b
^3*c*d*e^3 - 8*b^4*e^4)*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)])))/(315*b*c^2*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. \(1169\) vs. \(2(397)=794\).
time = 0.44, size = 1170, normalized size = 2.56 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(15*b^2*c^4*d^3*e^2*x-3*b^3*c^3*d^2*e^3*x+41*((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^3*d^3*e^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
^2*c^4*d^4*e-8*b*c^5*d^4*e*x-67*b^2*c^4*d*e^4*x^3+5*b*c^5*d^2*e^3*x^3-2*b^3*c^3*d*e^4*x^2+b^2*c^4*d^2*e^3*x^2+
13*b*c^5*d^3*e^2*x^2-15*((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*d*e^4+4*b^4*c^2*d*e^4*x-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^3*c^3*d^3*e^2+40*((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^4*d^4*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^
2*d^2*e^3+4*((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*d*e^4-4*((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^2*d^2*e^3-101*b*c^5*d*e^4*x^4-2*c^6*d^3*e^2*x^3+4*b^4*c^2*e^5*x^2-8*c^6*
d^4*e*x^2-35*c^6*e^5*x^6+8*((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*e^5-40*c^6*d*e^4*x^5-53*b^2*c^4*e^5*x^4+c^6*d^2*e^3*x^4+b^3*c^3*e^5*x^3-85*b
*c^5*e^5*x^5-16*((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^5*d^5+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^5*d^5)/c^4/x/(c*e*x^2+b*e*x+c*d*x+b*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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.93, size = 516, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{5} d^{5} - 40 \, b c^{4} d^{4} e + 22 \, b^{2} c^{3} d^{3} e^{2} + 7 \, b^{3} c^{2} d^{2} e^{3} + 11 \, b^{4} c d e^{4} - 8 \, b^{5} e^{5}\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^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 9 \, b^{2} c^{3} d^{2} e^{3} + 7 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\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^{5} d^{3} e^{2} + {\left (35 \, c^{5} x^{3} + 50 \, b c^{4} x^{2} + 3 \, b^{2} c^{3} x - 4 \, b^{3} c^{2}\right )} e^{5} + {\left (5 \, c^{5} d x^{2} + 11 \, b c^{4} d x + 3 \, b^{2} c^{3} d\right )} e^{4} - 3 \, {\left (2 \, c^{5} d^{2} x + 5 \, b c^{4} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{945 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/945*((16*c^5*d^5 - 40*b*c^4*d^4*e + 22*b^2*c^3*d^3*e^2 + 7*b^3*c^2*d^2*e^3 + 11*b^4*c*d*e^4 - 8*b^5*e^5)*sqr
t(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^5*d^4*e - 32*b*c^4*d^
3*e^2 + 9*b^2*c^3*d^2*e^3 + 7*b^3*c^2*d*e^4 - 8*b^4*c*e^5)*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, weierstra
ssPInverse(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^5*d^3*e^2 + (35*c^5*x^3 + 50*b*c^4*x^2 + 3*
b^2*c^3*x - 4*b^3*c^2)*e^5 + (5*c^5*d*x^2 + 11*b*c^4*d*x + 3*b^2*c^3*d)*e^4 - 3*(2*c^5*d^2*x + 5*b*c^4*d^2)*e^
3)*sqrt(c*x^2 + b*x)*sqrt(x*e + d))*e^(-5)/c^4

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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}} \sqrt {d + e x}\, 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)**(1/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*sqrt(d + e*x), 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)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)*sqrt(x*e + d), 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}\,\sqrt {d+e\,x} \,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)^(1/2),x)

[Out]

int((b*x + c*x^2)^(3/2)*(d + e*x)^(1/2), x)

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