3.1260 \(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=574 \[ -\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (2 b^3 e^3+3 b^2 c d e^2-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}+\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-4 b^3 e^3-9 b^2 c d e^2-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]

[Out]

-2/63*(-7*B*c*e*x-9*A*c*e-3*B*b*e+8*B*c*d)*(c*x^2+b*x)^(3/2)*(e*x+d)^(1/2)/c/e^2-2/315*(5*b*c*d*e*(-b*e+2*c*d)
*(-9*A*c*e-3*B*b*e+8*B*c*d)+(-2*b^2*e^2-3*b*c*d*e+8*c^2*d^2)*(9*A*c*e*(-b*e+2*c*d)-B*(-4*b^2*e^2-7*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)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1
/2)/c^(5/2)/e^5/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/315*d*(-b*e+c*d)*(9*A*c*e*(-b^2*e^2-16*b*c*d*e+16*c^2*d^2)
-B*(-4*b^3*e^3-9*b^2*c*d*e^2-120*b*c^2*d^2*e+128*c^3*d^3))*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^5/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/315*(9*A*c
*e*(b^2*e^2-11*b*c*d*e+8*c^2*d^2)-2*B*(2*b^3*e^3+3*b^2*c*d*e^2-42*b*c^2*d^2*e+32*c^3*d^3)-3*c*e*(9*A*c*e*(-b*e
+2*c*d)-B*(-4*b^2*e^2-7*b*c*d*e+16*c^2*d^2))*x)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^2/e^4

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Rubi [A]  time = 0.78, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {814, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (3 b^2 c d e^2+2 b^3 e^3-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}+\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-9 b^2 c d e^2-4 b^3 e^3-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(9*A*c*e*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2) - 2*B*(32*c^3*d^3 - 42*b*c^2*d^2*e + 3*b^2*c*d*e^
2 + 2*b^3*e^3) - 3*c*e*(9*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 7*b*c*d*e - 4*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/
(315*c^2*e^4) - (2*Sqrt[d + e*x]*(8*B*c*d - 3*b*B*e - 9*A*c*e - 7*B*c*e*x)*(b*x + c*x^2)^(3/2))/(63*c*e^2) - (
2*Sqrt[-b]*(5*b*c*d*e*(2*c*d - b*e)*(8*B*c*d - 3*b*B*e - 9*A*c*e) + (8*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2)*(9*A*c
*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 7*b*c*d*e - 4*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)])/(315*c^(5/2)*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*S
qrt[-b]*d*(c*d - b*e)*(9*A*c*e*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2) - B*(128*c^3*d^3 - 120*b*c^2*d^2*e - 9*b^2*
c*d*e^2 - 4*b^3*e^3))*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^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, 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 112

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 116

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

Rule 117

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 715

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 814

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 843

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 {(A+B x) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac {2 \int \frac {\left (-\frac {1}{2} b d (8 B c d-3 b B e-9 A c e)+\frac {1}{2} \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e^2}\\ &=\frac {2 \sqrt {d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^4}-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac {4 \int \frac {-\frac {1}{4} b d \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-84 b c^2 d^2 e+6 b^2 c d e^2+4 b^3 e^3\right )\right )-\frac {1}{4} \left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^4}-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac {\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^5}-\frac {\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{315 c^2 e^5}\\ &=\frac {2 \sqrt {d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^4}-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac {\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\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^5 \sqrt {b x+c x^2}}-\frac {\left (\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\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^5 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^4}-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac {\left (\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\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^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\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^5 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^4}-\frac {2 \sqrt {d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac {2 \sqrt {-b} \left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\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^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\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^5 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 4.55, size = 630, normalized size = 1.10 \[ -\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (B \left (4 b^3 e^3-3 b^2 c e^2 (e x-2 d)+b c^2 e \left (-84 d^2+61 d e x-50 e^2 x^2\right )+c^3 \left (64 d^3-48 d^2 e x+40 d e^2 x^2-35 e^3 x^3\right )\right )-9 A c e \left (b^2 e^2+b c e (8 e x-11 d)+c^2 \left (8 d^2-6 d e x+5 e^2 x^2\right )\right )\right )+\sqrt {\frac {b}{c}} \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) \left (9 A c e \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )+B \left (8 b^3 e^3+15 b^2 c d e^2+36 b c^2 d^2 e-64 c^3 d^3\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (18 A c e \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (8 b^4 e^4+11 b^3 c d e^3+27 b^2 c^2 d^2 e^2-184 b c^3 d^3 e+128 c^4 d^4\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (18 A c e \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (8 b^4 e^4+11 b^3 c d e^3+27 b^2 c^2 d^2 e^2-184 b c^3 d^3 e+128 c^4 d^4\right )\right )\right )\right )}{315 b c^2 e^5 x^2 (b+c x)^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-9*A*c*e*(b^2*e^2 + b*c*e*(-11*d + 8*e*x) + c^2*(8*d^2 - 6
*d*e*x + 5*e^2*x^2)) + B*(4*b^3*e^3 - 3*b^2*c*e^2*(-2*d + e*x) + b*c^2*e*(-84*d^2 + 61*d*e*x - 50*e^2*x^2) + c
^3*(64*d^3 - 48*d^2*e*x + 40*d*e^2*x^2 - 35*e^3*x^3))) + Sqrt[b/c]*(Sqrt[b/c]*(18*A*c*e*(8*c^3*d^3 - 12*b*c^2*
d^2*e + 2*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 184*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 + 8*
b^4*e^4))*(b + c*x)*(d + e*x) + I*b*e*(18*A*c*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3) - B*(12
8*c^4*d^4 - 184*b*c^3*d^3*e + 27*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)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(9*A*c*e*(8*c^2*d^2 - 5
*b*c*d*e - 2*b^2*e^2) + B*(-64*c^3*d^3 + 36*b*c^2*d^2*e + 15*b^2*c*d*e^2 + 8*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)])))/(315*b*c^2*e^5*x^2*(b + c*x)^2*Sq
rt[d + e*x])

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{3} + A b x + {\left (B b + A c\right )} x^{2}\right )} \sqrt {c x^{2} + b x}}{\sqrt {e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*c*x^3 + A*b*x + (B*b + A*c)*x^2)*sqrt(c*x^2 + b*x)/sqrt(e*x + d), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 2112, normalized size = 3.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((c*x+b)*x)^(1/2)*(e*x+d)^(1/2)*(-64*B*x*b*c^5*d^4*e+18*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)
*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*c^2*d*e^4+9*A*((c*x+b)/b)^(1/2)*(-(
e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*c^2*d*e^4+
135*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)
*b*e)^(1/2))*b^3*c^3*d^2*e^3-288*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF((
(c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^4*d^3*e^2+144*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)
*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^5*d^4*e+85*B*x^5*b*c^5*e^5-252*A*((
c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1
/2))*b^3*c^3*d^2*e^3+360*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/
b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^4*d^3*e^2-144*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c
*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^5*d^4*e-4*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*
e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^5*c*d*e^4-5*B*((c*x+b)
/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b
^4*c^2*d^2*e^3-111*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/
2),(1/(b*e-c*d)*b*e)^(1/2))*b^3*c^3*d^3*e^2+248*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1
/2)*EllipticF(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^4*d^4*e-3*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*
d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^5*c*d*e^4-16*B*((c*x+b)/b)
^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^4*
c^2*d^2*e^3+211*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),
(1/(b*e-c*d)*b*e)^(1/2))*b^3*c^3*d^3*e^2-312*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)
*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^2*c^4*d^4*e+18*A*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)
*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b^5*c*e^5+35*B*x^6*c^6*e^5+45*
A*x^5*c^6*e^5-128*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticF(((c*x+b)/b)^(1/2
),(1/(b*e-c*d)*b*e)^(1/2))*b*c^5*d^5+128*B*((c*x+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2))*b*c^5*d^5+8*B*x^4*c^6*d^2*e^3+81*A*x^3*b^2*c^4*e^5+18*A*x^3*
c^6*d^2*e^3-64*B*x^2*c^6*d^4*e-9*A*x^4*c^6*d*e^4+53*B*x^4*b^2*c^4*e^5-5*B*x^5*c^6*d*e^4+117*A*x^4*b*c^5*e^5-B*
x^3*b^3*c^3*e^5-16*B*x^3*c^6*d^3*e^2+9*A*x^2*b^3*c^3*e^5+72*A*x^2*c^6*d^3*e^2-4*B*x^2*b^4*c^2*e^5-4*B*x*b^4*c^
2*d*e^4-6*B*x*b^3*c^3*d^2*e^3+84*B*x*b^2*c^4*d^3*e^2+68*B*x^2*b*c^5*d^3*e^2+9*A*x*b^3*c^3*d*e^4-99*A*x*b^2*c^4
*d^2*e^3+17*B*x^2*b^2*c^4*d^2*e^3-18*A*x^2*b^2*c^4*d*e^4-81*A*x^2*b*c^5*d^2*e^3-7*B*x^2*b^3*c^3*d*e^4-16*B*x^4
*b*c^5*d*e^4-36*A*x^3*b*c^5*d*e^4-14*B*x^3*b^2*c^4*d*e^4+31*B*x^3*b*c^5*d^2*e^3+72*A*x*b*c^5*d^3*e^2-8*B*((c*x
+b)/b)^(1/2)*(-(e*x+d)/(b*e-c*d)*c)^(1/2)*(-1/b*c*x)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(1/(b*e-c*d)*b*e)^(1/2)
)*b^6*e^5)/c^4/e^5/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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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