3.1458 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a-c x^2)^3} \, dx\)

Optimal. Leaf size=396 \[ \frac{\sqrt{d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac{(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

[Out]

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*A*c*d^2 -
14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*(3*c*d^2 - 2*a*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(16*a^2*c^2*(a - c*x^2)
) + ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e
+ 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4)) - ((S
qrt[c]*d + Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*
Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4))

________________________________________________________________________________________

Rubi [A]  time = 0.877394, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {819, 827, 1166, 208} \[ \frac{\sqrt{d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac{(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*A*c*d^2 -
14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*(3*c*d^2 - 2*a*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(16*a^2*c^2*(a - c*x^2)
) + ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e
+ 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4)) - ((S
qrt[c]*d + Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*
Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4))

Rule 819

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 + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 827

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx &=\frac{(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (-6 A c d^2+a e (7 B d+5 A e)\right )-\frac{1}{2} e (A c d-7 a B e) x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac{(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{4} \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )+\frac{1}{4} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac{(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} e \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )-\frac{1}{4} d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )+\frac{1}{4} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{4 a^2 c^2}\\ &=\frac{(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac{\left (\left (\sqrt{c} d+\sqrt{a} e\right )^2 \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt{a} c d e+5 a \sqrt{c} e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{32 a^{5/2} c^2}+\frac{\left (\frac{1}{8} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )+\frac{-\frac{1}{2} c d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )-2 c \left (\frac{1}{4} e \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )-\frac{1}{4} d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )\right )}{4 \sqrt{a} \sqrt{c} e}\right ) \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a^2 c^2}\\ &=\frac{(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d+3 \sqrt{a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt{a} c d e+5 a \sqrt{c} e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac{\left (\sqrt{c} d+\sqrt{a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt{a} c d e+5 a \sqrt{c} e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}\\ \end{align*}

Mathematica [B]  time = 3.18848, size = 802, normalized size = 2.03 \[ \frac{\frac{2 a c^2 \left (c d^2-a e^2\right ) (-a A e+A c d x+a B (d-e x)) (d+e x)^{9/2}}{\left (a-c x^2\right )^2}+\frac{c^2 \left (6 A c^2 x d^3+a c e (-9 A d-7 B x d+4 A e x) d-a^2 e^2 (-10 B d+A e+3 B e x)\right ) (d+e x)^{9/2}}{2 \left (a-c x^2\right )}-\frac{\left (A \left (54 c^2 d^4+81 a c e^2 d^2+5 a^2 e^4\right )-7 a B d e \left (9 c d^2+11 a e^2\right )\right ) \left (15 \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}+2 \sqrt{a} \sqrt [4]{c} e \sqrt{d+e x} \left (15 a e^2+c \left (58 d^2+16 e x d+3 e^2 x^2\right )\right )-15 \left (\sqrt{c} d+\sqrt{a} e\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )\right )}{60 \sqrt{a} \sqrt [4]{c}}-\frac{\left (2 A c d \left (3 c d^2+2 a e^2\right )-a B e \left (7 c d^2+3 a e^2\right )\right ) \left (\left (\sqrt{c} d-\sqrt{a} e\right ) \left (15 c^{7/4} (d+e x)^{7/2}+7 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (3 c^{5/4} (d+e x)^{5/2}+5 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (\sqrt [4]{c} \sqrt{d+e x} \left (\sqrt{c} (4 d+e x)-3 \sqrt{a} e\right )-3 \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )\right )\right )\right )-\left (\sqrt{c} d+\sqrt{a} e\right ) \left (15 c^{7/4} (d+e x)^{7/2}+7 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (3 c^{5/4} (d+e x)^{5/2}+5 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (\sqrt [4]{c} \sqrt{d+e x} \left (3 \sqrt{a} e+\sqrt{c} (4 d+e x)\right )-3 \left (\sqrt{c} d+\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )\right )\right )\right )\right )}{60 \sqrt{a} c^{3/4}}}{8 a^2 c^2 \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*c^2*(c*d^2 - a*e^2)*(d + e*x)^(9/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*(d + e*x)
^(9/2)*(6*A*c^2*d^3*x + a*c*d*e*(-9*A*d - 7*B*d*x + 4*A*e*x) - a^2*e^2*(-10*B*d + A*e + 3*B*e*x)))/(2*(a - c*x
^2)) - ((-7*a*B*d*e*(9*c*d^2 + 11*a*e^2) + A*(54*c^2*d^4 + 81*a*c*d^2*e^2 + 5*a^2*e^4))*(2*Sqrt[a]*c^(1/4)*e*S
qrt[d + e*x]*(15*a*e^2 + c*(58*d^2 + 16*d*e*x + 3*e^2*x^2)) + 15*(Sqrt[c]*d - Sqrt[a]*e)^(7/2)*ArcTanh[(c^(1/4
)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - 15*(Sqrt[c]*d + Sqrt[a]*e)^(7/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(60*Sqrt[a]*c^(1/4)) - ((2*A*c*d*(3*c*d^2 + 2*a*e^2) - a*B*e*(7*c*d^2 + 3*a*
e^2))*((Sqrt[c]*d - Sqrt[a]*e)*(15*c^(7/4)*(d + e*x)^(7/2) + 7*(Sqrt[c]*d - Sqrt[a]*e)*(3*c^(5/4)*(d + e*x)^(5
/2) + 5*(Sqrt[c]*d - Sqrt[a]*e)*(c^(1/4)*Sqrt[d + e*x]*(-3*Sqrt[a]*e + Sqrt[c]*(4*d + e*x)) - 3*(Sqrt[c]*d - S
qrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]))) - (Sqrt[c]*d + Sqrt[a]*e)*(15*
c^(7/4)*(d + e*x)^(7/2) + 7*(Sqrt[c]*d + Sqrt[a]*e)*(3*c^(5/4)*(d + e*x)^(5/2) + 5*(Sqrt[c]*d + Sqrt[a]*e)*(c^
(1/4)*Sqrt[d + e*x]*(3*Sqrt[a]*e + Sqrt[c]*(4*d + e*x)) - 3*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])))))/(60*Sqrt[a]*c^(3/4)))/(8*a^2*c^2*(c*d^2 - a*e^2)^2)

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Maple [B]  time = 0.057, size = 1828, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/32*e^5/c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^
(1/2))*A-3/16*e/a^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*
A*d^3+1/2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*A*d+7/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*B*d^2-23/16
*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*A*d^2-21/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*B*d^3-7/8*e^5/(c*
e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*A*d+2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*A*d^3-7/16*e^6/(c*e^2*x^2-a*e^2
)^2/c^2*a*(e*x+d)^(3/2)*B+21/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*B*d^4-5/16*e^7/(c*e^2*x^2-a*e^2)^2*a/c
^2*(e*x+d)^(1/2)*A+e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*A*d^2-17/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)
*A*d^4-7/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*B*d^5-7/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)*B*d-7/8
*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*B*d^2+7/8*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*B*d^3+5/32*e^5/c/(a
*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A+3/16*
e/a^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^3-21/32*e^4
/c^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B+11/16*e^4/(c*
e^2*x^2-a*e^2)^2/c*(e*x+d)^(7/2)*B+9/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)*A+21/32*e^4/c^2/((c*d+(a*c*e^2
)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B+3/8*e/a^2*c/(a*c*e^2)^(1/2)/((-c*
d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^4-7/16*e^6/(c*e^2*x^2
-a*e^2)^2*a/c^2*(e*x+d)^(1/2)*B*d-7/16*e^2/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(
1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^3+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*A*d^6-9/8*e/(c*e^
2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(3/2)*A*d^5-19/32*e^3/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((
e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^2+3/8*e/a^2*c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1
/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^4+9/8*e/(c*e^2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(5/
2)*A*d^4-1/4*e^3/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*
A*d-7/32*e^2/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^
2+1/4*e^3/a/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d+7/
32*e^2/a/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^2-3/8
*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(7/2)*A*d^3-7/16*e^2/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^3-19/32*e^3/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2
))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^2+7/8*e^4/c/(a*c*e^2)^(1/2)/((-c*d+(a
*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d+7/8*e^4/c/(a*c*e^2)^(1/2)
/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 84.6927, size = 14866, normalized size = 37.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*
d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*
a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^
3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 53
2*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)
/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7
*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*
e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727
*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2
*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481
*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*
d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) + (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a
^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*
A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*
e^10 + (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^
4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^
4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 +
25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3
*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^
3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3
+ 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*
c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441
*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*
c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a
^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a
*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2
*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4
*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2
+ 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*
a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11
*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35
*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 -
108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (2
42501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B
*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B
^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) - (1260*A^2*B*a^
3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7
+ 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*
B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 + (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c
^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 -
2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*
a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (1944
81*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d
^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d
^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*
d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*
A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c
+ 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a
^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqr
t((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7
- a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^
2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B
^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (1
94481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d
^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(3024
0*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^
7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 80
19*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29
743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3
*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*
sqrt(e*x + d) + (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c
^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^
2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 - (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 1
3*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^
5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a
*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25
*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144
*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5
*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 20
70*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3
*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*
B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2
- 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - (a^2*c^4*x^4
- 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a
^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)
*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*
B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c +
25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7
*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c
)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3
*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7
- 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509
*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*
A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4
*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) - (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c
^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8
- 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 - (12*A*a^5*c^1
0*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35
*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(56
35*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^
12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2
)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d
^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e
^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^
3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^
3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a
^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^
6)/(a^5*c^5))) - 4*(4*B*a^2*c*d^3 + 11*A*a^2*c*d^2*e - 14*B*a^3*d*e^2 - 5*A*a^3*e^3 - (6*A*c^3*d^3 - 7*B*a*c^2
*d^2*e - 8*A*a*c^2*d*e^2 - 11*B*a^2*c*e^3)*x^3 + (A*a*c^2*d^2*e + 26*B*a^2*c*d*e^2 + 9*A*a^2*c*e^3)*x^2 + (10*
A*a*c^2*d^3 + 5*B*a^2*c*d^2*e + 4*A*a^2*c*d*e^2 - 7*B*a^3*e^3)*x)*sqrt(e*x + d))/(a^2*c^4*x^4 - 2*a^3*c^3*x^2
+ a^4*c^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out