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

Optimal. Leaf size=294 \[ \frac{\sqrt{d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac{(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

[Out]

((a*e + c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*c*d^2 - 5*a*e^2) + 2*c*d*(3*c*d
^2 - 2*a*e^2)*x))/(16*a^2*c^2*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e
+ 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((Sqrt[c]*d +
Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
+ Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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Rubi [A]  time = 0.524751, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {739, 819, 827, 1166, 208} \[ \frac{\sqrt{d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac{(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a - c*x^2)^3,x]

[Out]

((a*e + c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*c*d^2 - 5*a*e^2) + 2*c*d*(3*c*d
^2 - 2*a*e^2)*x))/(16*a^2*c^2*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e
+ 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((Sqrt[c]*d +
Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
+ Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

Rule 739

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

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{(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx &=\frac{(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}-\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (-6 c d^2+5 a e^2\right )-\frac{1}{2} c d e x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac{(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{4} \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac{1}{2} c d e \left (3 c d^2-4 a e^2\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac{(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} c d^2 e \left (3 c d^2-4 a e^2\right )+\frac{1}{4} e \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac{1}{2} c d e \left (3 c d^2-4 a e^2\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{(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\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 (12 c d^2-18 \sqrt{a} \sqrt{c} d e+5 a e^2\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^{3/2}}-\frac{\left (\left (\sqrt{c} d-\sqrt{a} e\right )^2 \left (12 c d^2+18 \sqrt{a} \sqrt{c} d e+5 a e^2\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^{3/2}}\\ &=\frac{(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\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 (12 c d^2+18 \sqrt{a} \sqrt{c} d e+5 a e^2\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^{9/4}}+\frac{\left (\sqrt{c} d+\sqrt{a} e\right )^{3/2} \left (12 c d^2-18 \sqrt{a} \sqrt{c} d e+5 a e^2\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^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.710775, size = 319, normalized size = 1.09 \[ \frac{\frac{2 \sqrt{a} \sqrt [4]{c} \sqrt{d+e x} \left (a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )-5 a^3 e^3+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-6 c^3 d^3 x^3\right )}{\left (a-c x^2\right )^2}-\sqrt{\sqrt{c} d-\sqrt{a} e} \left (-5 a^{3/2} e^3+6 \sqrt{a} c d^2 e-13 a \sqrt{c} d e^2+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )+\sqrt{\sqrt{a} e+\sqrt{c} d} \left (5 a^{3/2} e^3-6 \sqrt{a} c d^2 e-13 a \sqrt{c} d e^2+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(-5*a^3*e^3 - 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) + a^2*c
*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a - c*x^2)^2 - Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(12*c^(3/2)*d^3 + 6*Sqrt[a]*c*
d^2*e - 13*a*Sqrt[c]*d*e^2 - 5*a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + Sqr
t[Sqrt[c]*d + Sqrt[a]*e]*(12*c^(3/2)*d^3 - 6*Sqrt[a]*c*d^2*e - 13*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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Maple [B]  time = 0.234, size = 986, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^3/(c*e^2*x^2-a*e^2)^2*d/a*(e*x+d)^(7/2)-3/8*e/(c*e^2*x^2-a*e^2)^2*d^3/a^2*(e*x+d)^(7/2)*c+9/16*e^5/(c*e^
2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)-23/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*d^2+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2
*c*(e*x+d)^(5/2)*d^4-7/8*e^5/(c*e^2*x^2-a*e^2)^2*d/c*(e*x+d)^(3/2)+2*e^3/(c*e^2*x^2-a*e^2)^2*d^3/a*(e*x+d)^(3/
2)-9/8*e/(c*e^2*x^2-a*e^2)^2*d^5/a^2*c*(e*x+d)^(3/2)-5/16*e^7/(c*e^2*x^2-a*e^2)^2/c^2*a*(e*x+d)^(1/2)+e^5/(c*e
^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*d^2-17/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*d^4+3/8*e/(c*e^2*x^2-a*e^2)^
2*c/a^2*(e*x+d)^(1/2)*d^6+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))-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))*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)*arct
an((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^4+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))*d-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))*d^3+5/32*e^5/c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arcta
nh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))-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))*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))*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))*d+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))*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\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((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.28605, size = 3876, normalized size = 13.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*
a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((302
4*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d
) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2)*sqrt((
441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a
^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/
(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^
4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*
log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt
(e*x + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2
)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2
+ 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5
*c^9)))/(a^5*c^4))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*
c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^
5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^
13)*sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 - 13*a^6*c
^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2
*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^
14)/(a^5*c^9)))/(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 +
385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^
9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 62
5*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 -
13*a^6*c^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 4
20*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 62
5*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) + 4*(11*a^2*c*d^2*e - 5*a^3*e^3 - 2*(3*c^3*d^3 - 4*a*c^2*d*e^2)*x^3 + (a*c
^2*d^2*e + 9*a^2*c*e^3)*x^2 + 2*(5*a*c^2*d^3 + 2*a^2*c*d*e^2)*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((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((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

Timed out