3.665 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=553 \[ -\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (21 a^2 e^4+69 a c d^2 e^2+40 c^2 d^4\right )+d \left (9 a^2 e^4+49 a c d^2 e^2+32 c^2 d^4\right )\right )}{63 e^5 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac{16 \sqrt{-a} c^{5/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \left (33 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (7 a e^2+13 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{63 e^3 (d+e x)^{7/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

[Out]

(-8*c^2*(d*(32*c^2*d^4 + 49*a*c*d^2*e^2 + 9*a^2*e^4) + e*(40*c^2*d^4 + 69*a*c*d^
2*e^2 + 21*a^2*e^4)*x)*Sqrt[a + c*x^2])/(63*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2
)) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(13*c*d^2 + 7*a*e^2)*x)*(a + c*x^2)^(3/2))/
(63*e^3*(c*d^2 + a*e^2)*(d + e*x)^(7/2)) - (2*(a + c*x^2)^(5/2))/(9*e*(d + e*x)^
(9/2)) - (16*Sqrt[-a]*c^(5/2)*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*Sqrt[d
+ e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[
2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(63*e^6*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[
c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(5/2)*
d*(32*c*d^2 + 33*a*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)])/(63*e^6*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*
x^2])

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Rubi [A]  time = 1.45782, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (21 a^2 e^4+69 a c d^2 e^2+40 c^2 d^4\right )+d \left (9 a^2 e^4+49 a c d^2 e^2+32 c^2 d^4\right )\right )}{63 e^5 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac{16 \sqrt{-a} c^{5/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \left (33 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (7 a e^2+13 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{63 e^3 (d+e x)^{7/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-8*c^2*(d*(32*c^2*d^4 + 49*a*c*d^2*e^2 + 9*a^2*e^4) + e*(40*c^2*d^4 + 69*a*c*d^
2*e^2 + 21*a^2*e^4)*x)*Sqrt[a + c*x^2])/(63*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2
)) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(13*c*d^2 + 7*a*e^2)*x)*(a + c*x^2)^(3/2))/
(63*e^3*(c*d^2 + a*e^2)*(d + e*x)^(7/2)) - (2*(a + c*x^2)^(5/2))/(9*e*(d + e*x)^
(9/2)) - (16*Sqrt[-a]*c^(5/2)*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*Sqrt[d
+ e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[
2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(63*e^6*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[
c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*c^(5/2)*
d*(32*c*d^2 + 33*a*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)])/(63*e^6*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*
x^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((c*x**2+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Mathematica [C]  time = 9.09929, size = 913, normalized size = 1.65 \[ \frac{16 (d+e x)^{3/2} \left (\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (\frac{21 a^3 e^6}{(d+e x)^2}+3 a^2 c \left (\frac{26 d^2}{(d+e x)^2}-\frac{14 d}{d+e x}+7\right ) e^4+a c^2 d^2 \left (\frac{89 d^2}{(d+e x)^2}-\frac{114 d}{d+e x}+57\right ) e^2+32 c^3 d^4 \left (\frac{d}{d+e x}-1\right )^2\right )+\frac{\sqrt{c} \left (-32 i c^{5/2} d^5+32 \sqrt{a} c^2 e d^4-57 i a c^{3/2} e^2 d^3+57 a^{3/2} c e^3 d^2-21 i a^2 \sqrt{c} e^4 d+21 a^{5/2} e^5\right ) \sqrt{-\frac{d}{d+e x}-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}+1} \sqrt{-\frac{d}{d+e x}+\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}-\frac{\sqrt{a} \sqrt{c} e \left (32 c^2 d^4+8 i \sqrt{a} c^{3/2} e d^3+57 a c e^2 d^2+12 i a^{3/2} \sqrt{c} e^3 d+21 a^2 e^4\right ) \sqrt{-\frac{d}{d+e x}-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}+1} \sqrt{-\frac{d}{d+e x}+\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right ) c^2}{63 e^7 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (c d^2+a e^2\right )^2 \sqrt{\frac{c (d+e x)^2 \left (\frac{d}{d+e x}-1\right )^2}{e^2}+a}}+\sqrt{d+e x} \sqrt{c x^2+a} \left (-\frac{2 \left (193 c^2 d^4+330 a c e^2 d^2+105 a^2 e^4\right ) c^2}{63 e^5 \left (c d^2+a e^2\right )^2 (d+e x)}+\frac{4 d \left (61 c d^2+57 a e^2\right ) c^2}{63 e^5 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac{8 \left (22 c d^2+7 a e^2\right ) c}{63 e^5 (d+e x)^3}+\frac{76 d \left (c d^2+a e^2\right ) c}{63 e^5 (d+e x)^4}-\frac{2 \left (c d^2+a e^2\right )^2}{9 e^5 (d+e x)^5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

Sqrt[d + e*x]*Sqrt[a + c*x^2]*((-2*(c*d^2 + a*e^2)^2)/(9*e^5*(d + e*x)^5) + (76*
c*d*(c*d^2 + a*e^2))/(63*e^5*(d + e*x)^4) - (8*c*(22*c*d^2 + 7*a*e^2))/(63*e^5*(
d + e*x)^3) + (4*c^2*d*(61*c*d^2 + 57*a*e^2))/(63*e^5*(c*d^2 + a*e^2)*(d + e*x)^
2) - (2*c^2*(193*c^2*d^4 + 330*a*c*d^2*e^2 + 105*a^2*e^4))/(63*e^5*(c*d^2 + a*e^
2)^2*(d + e*x))) + (16*c^2*(d + e*x)^(3/2)*(Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*((2
1*a^3*e^6)/(d + e*x)^2 + a*c^2*d^2*e^2*(57 + (89*d^2)/(d + e*x)^2 - (114*d)/(d +
e*x)) + 3*a^2*c*e^4*(7 + (26*d^2)/(d + e*x)^2 - (14*d)/(d + e*x)) + 32*c^3*d^4*
(-1 + d/(d + e*x))^2) + (Sqrt[c]*((-32*I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^4*e - (
57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a
^(5/2)*e^5)*Sqrt[1 - d/(d + e*x) - (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*Sqrt[1 - d
/(d + e*x) + (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*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*Sq
rt[a]*e)])/Sqrt[d + e*x] - (Sqrt[a]*Sqrt[c]*e*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2
)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^3 + 21*a^2*e^4)*Sqrt[1 - d
/(d + e*x) - (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*Sqrt[1 - d/(d + e*x) + (I*Sqrt[a
]*e)/(Sqrt[c]*(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/S
qrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*
x]))/(63*e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)^2*Sqrt[a + (c*(d +
e*x)^2*(-1 + d/(d + e*x))^2)/e^2])

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Maple [B]  time = 0.109, size = 8244, normalized size = 14.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a}}{{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)/((e^5*x^5 + 5*d*e^4*x^4 + 1
0*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError