### 3.1723 $$\int \frac{(d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=250 $-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}$

[Out]

(-35*e^3*Sqrt[d + e*x])/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*x)^(3/2))/(96*b^3*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d +
e*x)^(7/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x]
)/Sqrt[b*d - a*e]])/(64*b^(9/2)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.11792, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {646, 47, 63, 208} $-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-35*e^3*Sqrt[d + e*x])/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(d + e*x)^(3/2))/(96*b^3*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(d + e*x)^(5/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d +
e*x)^(7/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x]
)/Sqrt[b*d - a*e]])/(64*b^(9/2)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{7/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 b^2 e \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^3 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{128 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^3 \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{b d-a e} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.223078, size = 182, normalized size = 0.73 $\frac{\frac{105 e^4 (a+b x)^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{\sqrt{a e-b d}}-\sqrt{b} \sqrt{d+e x} \left (35 a^2 b e^2 (2 d+11 e x)+105 a^3 e^3+7 a b^2 e \left (8 d^2+36 d e x+73 e^2 x^2\right )+b^3 \left (200 d^2 e x+48 d^3+326 d e^2 x^2+279 e^3 x^3\right )\right )}{192 b^{9/2} (a+b x)^3 \sqrt{(a+b x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-(Sqrt[b]*Sqrt[d + e*x]*(105*a^3*e^3 + 35*a^2*b*e^2*(2*d + 11*e*x) + 7*a*b^2*e*(8*d^2 + 36*d*e*x + 73*e^2*x^2
) + b^3*(48*d^3 + 200*d^2*e*x + 326*d*e^2*x^2 + 279*e^3*x^3))) + (105*e^4*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/Sqrt[-(b*d) + a*e])/(192*b^(9/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.277, size = 467, normalized size = 1.9 \begin{align*} -{\frac{bx+a}{192\,{b}^{4}} \left ( -105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{4}{b}^{4}{e}^{4}-420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{3}a{b}^{3}{e}^{4}+279\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{7/2}{b}^{3}-630\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+511\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}a{b}^{2}e-511\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}{b}^{3}d-420\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) x{a}^{3}b{e}^{4}+385\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-770\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}a{b}^{2}de+385\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}-105\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{4}{e}^{4}+105\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{3}{e}^{3}-315\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}bd{e}^{2}+315\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}a{b}^{2}{d}^{2}e-105\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/192*(-105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*x^4*b^4*e^4-420*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^
(1/2))*x^3*a*b^3*e^4+279*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^3-630*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))
*x^2*a^2*b^2*e^4+511*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^2*e-511*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^3*d-420
*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*x*a^3*b*e^4+385*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-770*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+385*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^3*d^2-105*arctan(b*(e*x+d)^
(1/2)/((a*e-b*d)*b)^(1/2))*a^4*e^4+105*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*e^3-315*((a*e-b*d)*b)^(1/2)*(e*x+
d)^(1/2)*a^2*b*d*e^2+315*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e-105*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b
^3*d^3)*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^4/((b*x+a)^2)^(5/2)

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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 (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

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Fricas [B]  time = 1.70959, size = 1602, normalized size = 6.41 \begin{align*} \left [\frac{105 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (48 \, b^{5} d^{4} + 8 \, a b^{4} d^{3} e + 14 \, a^{2} b^{3} d^{2} e^{2} + 35 \, a^{3} b^{2} d e^{3} - 105 \, a^{4} b e^{4} + 279 \,{\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} +{\left (326 \, b^{5} d^{2} e^{2} + 185 \, a b^{4} d e^{3} - 511 \, a^{2} b^{3} e^{4}\right )} x^{2} +{\left (200 \, b^{5} d^{3} e + 52 \, a b^{4} d^{2} e^{2} + 133 \, a^{2} b^{3} d e^{3} - 385 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{384 \,{\left (a^{4} b^{6} d - a^{5} b^{5} e +{\left (b^{10} d - a b^{9} e\right )} x^{4} + 4 \,{\left (a b^{9} d - a^{2} b^{8} e\right )} x^{3} + 6 \,{\left (a^{2} b^{8} d - a^{3} b^{7} e\right )} x^{2} + 4 \,{\left (a^{3} b^{7} d - a^{4} b^{6} e\right )} x\right )}}, \frac{105 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (48 \, b^{5} d^{4} + 8 \, a b^{4} d^{3} e + 14 \, a^{2} b^{3} d^{2} e^{2} + 35 \, a^{3} b^{2} d e^{3} - 105 \, a^{4} b e^{4} + 279 \,{\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} +{\left (326 \, b^{5} d^{2} e^{2} + 185 \, a b^{4} d e^{3} - 511 \, a^{2} b^{3} e^{4}\right )} x^{2} +{\left (200 \, b^{5} d^{3} e + 52 \, a b^{4} d^{2} e^{2} + 133 \, a^{2} b^{3} d e^{3} - 385 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{192 \,{\left (a^{4} b^{6} d - a^{5} b^{5} e +{\left (b^{10} d - a b^{9} e\right )} x^{4} + 4 \,{\left (a b^{9} d - a^{2} b^{8} e\right )} x^{3} + 6 \,{\left (a^{2} b^{8} d - a^{3} b^{7} e\right )} x^{2} + 4 \,{\left (a^{3} b^{7} d - a^{4} b^{6} e\right )} x\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(b^2*d - a*b*e)*
log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(48*b^5*d^4 + 8*a*b^4*d^3*e + 1
4*a^2*b^3*d^2*e^2 + 35*a^3*b^2*d*e^3 - 105*a^4*b*e^4 + 279*(b^5*d*e^3 - a*b^4*e^4)*x^3 + (326*b^5*d^2*e^2 + 18
5*a*b^4*d*e^3 - 511*a^2*b^3*e^4)*x^2 + (200*b^5*d^3*e + 52*a*b^4*d^2*e^2 + 133*a^2*b^3*d*e^3 - 385*a^3*b^2*e^4
)*x)*sqrt(e*x + d))/(a^4*b^6*d - a^5*b^5*e + (b^10*d - a*b^9*e)*x^4 + 4*(a*b^9*d - a^2*b^8*e)*x^3 + 6*(a^2*b^8
*d - a^3*b^7*e)*x^2 + 4*(a^3*b^7*d - a^4*b^6*e)*x), 1/192*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*
x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) -
(48*b^5*d^4 + 8*a*b^4*d^3*e + 14*a^2*b^3*d^2*e^2 + 35*a^3*b^2*d*e^3 - 105*a^4*b*e^4 + 279*(b^5*d*e^3 - a*b^4*
e^4)*x^3 + (326*b^5*d^2*e^2 + 185*a*b^4*d*e^3 - 511*a^2*b^3*e^4)*x^2 + (200*b^5*d^3*e + 52*a*b^4*d^2*e^2 + 133
*a^2*b^3*d*e^3 - 385*a^3*b^2*e^4)*x)*sqrt(e*x + d))/(a^4*b^6*d - a^5*b^5*e + (b^10*d - a*b^9*e)*x^4 + 4*(a*b^9
*d - a^2*b^8*e)*x^3 + 6*(a^2*b^8*d - a^3*b^7*e)*x^2 + 4*(a^3*b^7*d - a^4*b^6*e)*x)]

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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)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20095, size = 387, normalized size = 1.55 \begin{align*} \frac{35 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \, \sqrt{-b^{2} d + a b e} b^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{279 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 511 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} - 105 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 511 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 770 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} + 315 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} - 315 \, \sqrt{x e + d} a^{2} b d e^{6} + 105 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

35/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/(sqrt(-b^2*d + a*b*e)*b^4*sgn((x*e + d)*b*e - b*d*e + a
*e^2)) - 1/192*(279*(x*e + d)^(7/2)*b^3*e^4 - 511*(x*e + d)^(5/2)*b^3*d*e^4 + 385*(x*e + d)^(3/2)*b^3*d^2*e^4
- 105*sqrt(x*e + d)*b^3*d^3*e^4 + 511*(x*e + d)^(5/2)*a*b^2*e^5 - 770*(x*e + d)^(3/2)*a*b^2*d*e^5 + 315*sqrt(x
*e + d)*a*b^2*d^2*e^5 + 385*(x*e + d)^(3/2)*a^2*b*e^6 - 315*sqrt(x*e + d)*a^2*b*d*e^6 + 105*sqrt(x*e + d)*a^3*
e^7)/(((x*e + d)*b - b*d + a*e)^4*b^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))