3.607 \(\int \frac{(d+e x)^{5/2}}{a+c x^2} \, dx\)

Optimal. Leaf size=781 \[ \frac{e \left (\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left (\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left (-\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{e \left (-\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{4 d e \sqrt{d+e x}}{c} \]

[Out]

(4*d*e*Sqrt[d + e*x])/c + (2*e*(d + e*x)^(3/2))/(3*c) - (e*(2*c^(3/2)*d^3 + 2*a*
Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d +
Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2
+ a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^
2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2
+ a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c
]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[
2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*
x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]
]) - (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^
2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^
2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*S
qrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 6.13866, antiderivative size = 781, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{e \left (\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left (\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left (-\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{e \left (-\left (3 c d^2-a e^2\right ) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{4 d e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

(4*d*e*Sqrt[d + e*x])/c + (2*e*(d + e*x)^(3/2))/(3*c) - (e*(2*c^(3/2)*d^3 + 2*a*
Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d +
Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2
+ a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^
2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2
+ a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c
]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[
2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*
x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]
]) - (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^
2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^
2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*S
qrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.243553, size = 217, normalized size = 0.28 \[ -\frac{i \left (\sqrt{c} d-i \sqrt{a} e\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c^{3/2} \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i \left (\sqrt{c} d+i \sqrt{a} e\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c^{3/2} \sqrt{c d+i \sqrt{a} \sqrt{c} e}}+\frac{2 e \sqrt{d+e x} (7 d+e x)}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[d + e*x]*(7*d + e*x))/(3*c) - (I*(Sqrt[c]*d - I*Sqrt[a]*e)^3*ArcTanh[(
Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^(3/2)*Sqrt[c
*d - I*Sqrt[a]*Sqrt[c]*e]) + (I*(Sqrt[c]*d + I*Sqrt[a]*e)^3*ArcTanh[(Sqrt[c]*Sqr
t[d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^(3/2)*Sqrt[c*d + I*Sqrt
[a]*Sqrt[c]*e])

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Maple [B]  time = 0.189, size = 3931, normalized size = 5. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+a),x)

[Out]

2/3*e*(e*x+d)^(3/2)/c+1/2*e/c^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c
*d^2))^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^
(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c
*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*
c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-3/4/c^(3/2)/a/e*ln((e*x+d)*c^(1/2)+(2*(c*(a*e
^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2
*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+3/4/c^(3/2)/a/e*ln(-(e*x+d)
*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/
2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+4*d*e*(e
*x+d)^(1/2)/c+1/4*e/c^(5/2)*ln((e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)
^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2
)*(a*c*e^2+c^2*d^2)^(1/2)+1/4*e/c^(3/2)*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))
^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2
)+2*c*d)^(1/2)*d-1/4*e/c^(5/2)*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*
c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^
(1/2)*(a*c*e^2+c^2*d^2)^(1/2)+4*e/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c
*d^2))^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^
(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c
*d)^(1/2))*(a*e^2+c*d^2)^(1/2)*d-3/4/c^(1/2)/a/e*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e^
2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*
d^2)^(1/2)+2*c*d)^(1/2)*d^3+3/4/c^(1/2)/a/e*ln((e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^
2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(
1/2)+2*c*d)^(1/2)*d^3-4*e/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(
1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c
*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)
)*(a*e^2+c*d^2)^(1/2)*d+1/c^2/a/e/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d
^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/
2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)
^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+
c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/c^2/a/e/(4*(a*e^2+c*d^2)
^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)
^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2
*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(
a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/
2)*d-1/2/c^2/a/e*ln((e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x
+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*
c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+1/2/c^2/a/e*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e^
2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1
/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+3/2/c^(1/2
)/a/e/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arct
an((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+
c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2)
)^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3-3/2/c^(1/2)/a/e
/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2
*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)
^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2
)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3-3/2/c^(3/2)/a/e/(4*(a
*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1
/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2
)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c
*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2-1/
2/c/a/e*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2
)-(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*d^2+1/2/c/a/e*ln((e*x+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x
+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*
c*d)^(1/2)*d^2+1/2*e/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^
(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*
c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2
))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*d-1/2*e/c^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*
d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)
)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(
a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^
2+c^2*d^2)^(1/2)-1/2*e/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2)
)^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)
+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(
1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^
(1/2)*d+3/2/c^(3/2)/a/e/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)
-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^
(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2
*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a
*c*e^2+c^2*d^2)^(1/2)*d^2+1/c/a/e/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d
^2))^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1
/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d
)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2
+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2-1/c/a/e/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*
e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^
2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)
-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a
*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2-1/4*e/c^(3/2)*ln((e*x+d)*c^(1/2)+(2*(c*(a
*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c
^2*d^2)^(1/2)+2*c*d)^(1/2)*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.297565, size = 2215, normalized size = 2.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + a),x, algorithm="fricas")

[Out]

-1/6*(3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^
8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(
a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*
e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 +
(a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^
2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e
^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2
*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt(-(c^2*d^5
- 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^
4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5
*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10
*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 - a^2*c^5*e^2
)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2
*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3
*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*
e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) + 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^
2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6
- 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*
d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 - 20*a^2*
c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2
- 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)
))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 -
100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7))
)/(a*c^3))) - 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(2
5*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4
*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^
7 + a^4*e^9)*sqrt(e*x + d) - (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*
d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 11
0*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a
*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110
*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 4*(e^2*x +
7*d*e)*sqrt(e*x + d))/c

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Sympy [A]  time = 116.248, size = 418, normalized size = 0.54 \[ - \frac{4 a d e^{3} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )}}{c} - \frac{2 a e^{3} \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )}}{c} - 4 d^{3} e \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )} + 6 d^{2} e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} + \frac{4 d e \sqrt{d + e x}}{c} + \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

-4*a*d*e**3*RootSum(_t**4*(256*a**3*c*e**6 + 256*a**2*c**2*d**2*e**4) + 32*_t**2
*a*c*d*e**2 + 1, Lambda(_t, _t*log(-64*_t**3*a**2*c*d*e**4 - 64*_t**3*a*c**2*d**
3*e**2 + 4*_t*a*e**2 - 4*_t*c*d**2 + sqrt(d + e*x))))/c - 2*a*e**3*RootSum(256*_
t**4*a**2*c**3*e**4 + 32*_t**2*a*c**2*d*e**2 + a*e**2 + c*d**2, Lambda(_t, _t*lo
g(64*_t**3*a*c**2*e**2 + 4*_t*c*d + sqrt(d + e*x))))/c - 4*d**3*e*RootSum(_t**4*
(256*a**3*c*e**6 + 256*a**2*c**2*d**2*e**4) + 32*_t**2*a*c*d*e**2 + 1, Lambda(_t
, _t*log(-64*_t**3*a**2*c*d*e**4 - 64*_t**3*a*c**2*d**3*e**2 + 4*_t*a*e**2 - 4*_
t*c*d**2 + sqrt(d + e*x)))) + 6*d**2*e*RootSum(256*_t**4*a**2*c**3*e**4 + 32*_t*
*2*a*c**2*d*e**2 + a*e**2 + c*d**2, Lambda(_t, _t*log(64*_t**3*a*c**2*e**2 + 4*_
t*c*d + sqrt(d + e*x)))) + 4*d*e*sqrt(d + e*x)/c + 2*e*(d + e*x)**(3/2)/(3*c)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + a),x, algorithm="giac")

[Out]

Exception raised: TypeError