3.1479 \(\int \frac{(A+B x) (a+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=541 \[ -\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-12 A c d e+32 B c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )} \]

[Out]

(4*c*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*c*d*e + 5*a*B*e^2)*x)*Sqrt[a +
c*x^2])/(15*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*(2*B*(4*c*d^3 + a*d*e^2) - 3*A*(c*d^2*e - a*e^3) + e*(11*
B*c*d^2 - 6*A*c*d*e + 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) + (8*Sqrt[-a]*
c^(3/2)*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*(c*d^2 + a*e^2)*Sqr
t[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^2 - 12*A*c*d*
e + 5*a*B*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1
- (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.471443, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {811, 813, 844, 719, 424, 419} \[ \frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-12 A c d e+32 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*c*d*e + 5*a*B*e^2)*x)*Sqrt[a +
c*x^2])/(15*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*(2*B*(4*c*d^3 + a*d*e^2) - 3*A*(c*d^2*e - a*e^3) + e*(11*
B*c*d^2 - 6*A*c*d*e + 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) + (8*Sqrt[-a]*
c^(3/2)*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*(c*d^2 + a*e^2)*Sqr
t[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^2 - 12*A*c*d*
e + 5*a*B*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1
- (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac{2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{\left (3 a c e (B d-A e)-c \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2 \left (c d^2+a e^2\right )}\\ &=\frac{4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac{4 \int \frac{a c e \left (8 B c d^2-3 A c d e+5 a B e^2\right )-c^2 \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 e^4 \left (c d^2+a e^2\right )}\\ &=\frac{4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac{\left (4 c \left (32 B c d^2-12 A c d e+5 a B e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 e^5}-\frac{\left (4 c^2 \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{15 e^5 \left (c d^2+a e^2\right )}\\ &=\frac{4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{\left (8 a c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{15 \sqrt{-a} e^5 \left (c d^2+a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (8 a \sqrt{c} \left (32 B c d^2-12 A c d e+5 a B e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{15 \sqrt{-a} e^5 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{4 c \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3+e \left (8 B c d^2-3 A c d e+5 a B e^2\right ) x\right ) \sqrt{a+c x^2}}{15 e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (2 B \left (4 c d^3+a d e^2\right )-3 A \left (c d^2 e-a e^3\right )+e \left (11 B c d^2-6 A c d e+5 a B e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{15 e^2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}+\frac{8 \sqrt{-a} c^{3/2} \left (32 B c d^3-12 A c d^2 e+29 a B d e^2-9 a A e^3\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{8 \sqrt{-a} \sqrt{c} \left (32 B c d^2-12 A c d e+5 a B e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 5.6069, size = 705, normalized size = 1.3 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (\frac{c \left (-21 a A e^3+61 a B d e^2-33 A c d^2 e+73 B c d^3\right )}{(d+e x) \left (a e^2+c d^2\right )}+\frac{-5 a B e^2+12 A c d e-17 B c d^2}{(d+e x)^2}+\frac{3 \left (a e^2+c d^2\right ) (B d-A e)}{(d+e x)^3}+5 B c\right )}{e^4}-\frac{8 c \left (-\sqrt{a} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (9 i \sqrt{a} A \sqrt{c} e^2-24 i \sqrt{a} B \sqrt{c} d e+5 a B e^2-12 A c d e+32 B c d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )+i \sqrt{c} (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (9 a A e^3-29 a B d e^2+12 A c d^2 e-32 B c d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e^6 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )}\right )}{15 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(5*B*c + (3*(B*d - A*e)*(c*d^2 + a*e^2))/(d + e*x)^3 + (-17*B*c*d^2 + 12*A*c*d*
e - 5*a*B*e^2)/(d + e*x)^2 + (c*(73*B*c*d^3 - 33*A*c*d^2*e + 61*a*B*d*e^2 - 21*a*A*e^3))/((c*d^2 + a*e^2)*(d +
e*x))))/e^4 - (8*c*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^
3)*(a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*d + I*Sqrt[a]*e)*(-32*B*c*d^3 + 12*A*c*d^2*e - 29*a*B*d*e^2 + 9*a*A*e^3)*S
qrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*
EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*
Sqrt[a]*e)] - Sqrt[a]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(32*B*c*d^2 - (24*I)*Sqrt[a]*B*Sqrt[c]*d*e - 12*A*c*d*e + 5*
a*B*e^2 + (9*I)*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqr
t[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (
Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^6*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)*(d
+ e*x))))/(15*Sqrt[a + c*x^2])

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Maple [B]  time = 0.078, size = 7383, normalized size = 13.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

integral((B*c*x^3 + A*c*x^2 + B*a*x + A*a)*sqrt(c*x^2 + a)*sqrt(e*x + d)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^
2 + 4*d^3*e*x + d^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Integral((A + B*x)*(a + c*x**2)**(3/2)/(d + e*x)**(7/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(7/2), x)