3.401 \(\int \frac{(b x+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=457 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3*c*e*(32*c^2*d^2 - 32*b*c*d*e
+ b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/
2))/(63*e^3) - (2*(b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (4*Sqrt[-b]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^
2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e
)*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.588482, antiderivative size = 457, normalized size of antiderivative = 1., 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 = {732, 814, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3*c*e*(32*c^2*d^2 - 32*b*c*d*e
+ b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/
2))/(63*e^3) - (2*(b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (4*Sqrt[-b]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^
2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e
)*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 732

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 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]

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 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 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 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 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)])

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}+\frac{5 \int \frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{e}\\ &=-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}-\frac{10 \int \frac{\left (-\frac{1}{2} b c d (16 c d-15 b e)-\frac{1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e^3}\\ &=-\frac{2 \sqrt{d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{63 c e^5}-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}+\frac{4 \int \frac{\frac{1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac{1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{63 c^2 e^5}\\ &=-\frac{2 \sqrt{d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{63 c e^5}-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}-\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{63 c e^6}+\frac{\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{63 c e^6}\\ &=-\frac{2 \sqrt{d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{63 c e^5}-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}-\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 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}{63 c e^6 \sqrt{b x+c x^2}}+\frac{\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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}{63 c e^6 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{63 c e^5}-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}+\frac{\left (2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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}{63 c e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 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}{63 c e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{63 c e^5}-\frac{10 \sqrt{d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}}+\frac{4 \sqrt{-b} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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 )}{63 c^{3/2} e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 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 )}{63 c^{3/2} e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.41372, size = 498, normalized size = 1.09 \[ \frac{2 (x (b+c x))^{5/2} \left (i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-159 b^2 c^2 d^2 e^2+13 b^3 c d e^3+2 b^4 e^4+272 b c^3 d^3 e-128 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-e \sqrt{x} (b+c x) \left (3 b^2 c e^2 \left (37 d^2+11 d e x-5 e^2 x^2\right )-b^3 e^3 (d+e x)-b c^2 e \left (64 d^2 e x+240 d^3-31 d e^2 x^2+19 e^3 x^3\right )+c^3 \left (-16 d^2 e^2 x^2+32 d^3 e x+128 d^4+10 d e^3 x^3-7 e^4 x^4\right )\right )+\frac{2 (b+c x) (d+e x) \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )}{c \sqrt{x}}-2 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4+256 b c^3 d^3 e-128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*(b
+ c*x)*(d + e*x))/(c*Sqrt[x]) - e*Sqrt[x]*(b + c*x)*(-(b^3*e^3*(d + e*x)) + 3*b^2*c*e^2*(37*d^2 + 11*d*e*x - 5
*e^2*x^2) - b*c^2*e*(240*d^3 + 64*d^2*e*x - 31*d*e^2*x^2 + 19*e^3*x^3) + c^3*(128*d^4 + 32*d^3*e*x - 16*d^2*e^
2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)) - (2*I)*Sqrt[b/c]*e*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 +
7*b^3*c*d*e^3 + b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/
(b*e)] + I*Sqrt[b/c]*e*(-128*c^4*d^4 + 272*b*c^3*d^3*e - 159*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 + 2*b^4*e^4)*Sqr
t[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(63*c*e^6*x^(5/2)*(b
+ c*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.29, size = 1170, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x)

[Out]

2/63*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(26*x^5*b*c^5*e^5-10*x^5*c^6*d*e^4+34*x^4*b^2*c^4*e^5+16*x^4*c^6*d^2*e^3+
16*x^3*b^3*c^3*e^5-32*x^3*c^6*d^3*e^2+x^2*b^4*c^2*e^5-128*x^2*c^6*d^4*e-256*(-(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))*((c*x+b)/b)^(1/2)*b*c^5*d^5+256*(-(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))*((c*x+b)/b)^(1/2)*b*c^5*d^5+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))*((c*x+b)/b)^(1/2)
*b^6*e^5+782*(-(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))*((
c*x+b)/b)^(1/2)*b^3*c^3*d^3*e^2-284*(-(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))*((c*x+b)/b)^(1/2)*b^4*c^2*d^2*e^3+640*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*((c*x+b)/b)^(1/2)*b^2*c^4*d^4*e+12*(-(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))*((c*x+b)/b)^(1/2)*b^5*c*d*e^4-510*(-(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))*((c*x+b)/b)^(1/2)*b^3*c^3*d
^3*e^2+125*(-(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))*((c*
x+b)/b)^(1/2)*b^4*c^2*d^2*e^3+(-(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))*((c*x+b)/b)^(1/2)*b^5*c*d*e^4-41*x^4*b*c^5*d*e^4+7*x^6*c^6*e^5-64*x^3*b^2*c^4*d*e^4+80*x^3*b*c^
5*d^2*e^3-32*x^2*b^3*c^3*d*e^4-47*x^2*b^2*c^4*d^2*e^3+208*x^2*b*c^5*d^3*e^2+x*b^4*c^2*d*e^4-111*x*b^3*c^3*d^2*
e^3+240*x*b^2*c^4*d^3*e^2-128*x*b*c^5*d^4*e-768*(-(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))*((c*x+b)/b)^(1/2)*b^2*c^4*d^4*e)/c^3/e^6/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(e^2*x^2 + 2*d*e*x + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2), x)