3.1479 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=541 \[ \frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-12 A c d e+32 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )-3 A \left (c d^2 e-a e^3\right )+2 B \left (a d e^2+4 c d^3\right )\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}+\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2+c d^2\right )} \]

[Out]

(4*c*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*
c*d*e + 5*a*B*e^2)*x)*Sqrt[a + c*x^2])/(15*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]) -
(2*(2*B*(4*c*d^3 + a*d*e^2) - 3*A*(c*d^2*e - a*e^3) + e*(11*B*c*d^2 - 6*A*c*d*e
+ 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) + (8
*Sqrt[-a]*c^(3/2)*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*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)])/(15*e^5*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*Sqrt[c]*(32
*B*c*d^2 - 12*A*c*d*e + 5*a*B*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)])/(15*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.24543, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-12 A c d e+32 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 e^5 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \left (a+c x^2\right )^{3/2} \left (e x \left (5 a B e^2-6 A c d e+11 B c d^2\right )+3 a A e^3+2 a B d e^2-3 A c d^2 e+8 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}+\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a B e^2-3 A c d e+8 B c d^2\right )-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3 + e*(8*B*c*d^2 - 3*A*
c*d*e + 5*a*B*e^2)*x)*Sqrt[a + c*x^2])/(15*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]) -
(2*(8*B*c*d^3 - 3*A*c*d^2*e + 2*a*B*d*e^2 + 3*a*A*e^3 + e*(11*B*c*d^2 - 6*A*c*d*
e + 5*a*B*e^2)*x)*(a + c*x^2)^(3/2))/(15*e^2*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) +
(8*Sqrt[-a]*c^(3/2)*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqr
t[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*e^5*(c*d^2 + a*e^2)*Sqrt[(Sqrt[
c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*Sqrt[c]*(
32*B*c*d^2 - 12*A*c*d*e + 5*a*B*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)])/(15*e^5*Sqrt[d + e*x]*Sqrt[a + c*x^2
])

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Rubi in Sympy [A]  time = 178.331, size = 547, normalized size = 1.01 \[ - \frac{8 c^{\frac{3}{2}} \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (9 A a e^{3} + 12 A c d^{2} e - 29 B a d e^{2} - 32 B c d^{3}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 e^{5} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{8 \sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (- 12 A c d e + 5 B a e^{2} + 32 B c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 e^{5} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{4 c \sqrt{a + c x^{2}} \left (9 A a e^{3} + 12 A c d^{2} e - 29 B a d e^{2} - 32 B c d^{3} - e x \left (5 B a e^{2} - c d \left (3 A e - 8 B d\right )\right )\right )}{15 e^{4} \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )} - \frac{4 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (\frac{3 A a e^{3}}{2} - \frac{3 A c d^{2} e}{2} + B a d e^{2} + 4 B c d^{3} + \frac{e x \left (- 6 A c d e + 5 B a e^{2} + 11 B c d^{2}\right )}{2}\right )}{15 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

-8*c**(3/2)*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(9*A*a*e**3 + 12*A*c*d**2*
e - 29*B*a*d*e**2 - 32*B*c*d**3)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) +
1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(15*e**5*sqrt(sqrt(c)*sqrt(-a)*(-d - e*
x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)*(a*e**2 + c*d**2)) - 8*sqrt(c)*s
qrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*
x**2/a)*(-12*A*c*d*e + 5*B*a*e**2 + 32*B*c*d**2)*elliptic_f(asin(sqrt(-sqrt(c)*x
/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(15*e**5*sqrt(a + c*x**
2)*sqrt(d + e*x)) - 4*c*sqrt(a + c*x**2)*(9*A*a*e**3 + 12*A*c*d**2*e - 29*B*a*d*
e**2 - 32*B*c*d**3 - e*x*(5*B*a*e**2 - c*d*(3*A*e - 8*B*d)))/(15*e**4*sqrt(d + e
*x)*(a*e**2 + c*d**2)) - 4*(a + c*x**2)**(3/2)*(3*A*a*e**3/2 - 3*A*c*d**2*e/2 +
B*a*d*e**2 + 4*B*c*d**3 + e*x*(-6*A*c*d*e + 5*B*a*e**2 + 11*B*c*d**2)/2)/(15*e**
2*(d + e*x)**(5/2)*(a*e**2 + c*d**2))

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Mathematica [C]  time = 10.1805, size = 789, normalized size = 1.46 \[ \sqrt{a+c x^2} \sqrt{d+e x} \left (\frac{2 \left (-5 a B e^2+12 A c d e-17 B c d^2\right )}{15 e^4 (d+e x)^2}-\frac{2 \left (a e^2+c d^2\right ) (A e-B d)}{5 e^4 (d+e x)^3}-\frac{2 c \left (21 a A e^3-61 a B d e^2+33 A c d^2 e-73 B c d^3\right )}{15 e^4 (d+e x) \left (a e^2+c d^2\right )}+\frac{2 B c}{3 e^4}\right )-\frac{8 c (d+e x)^{3/2} \left (\frac{\sqrt{a} e \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \left (-9 i \sqrt{a} A \sqrt{c} e^2+24 i \sqrt{a} B \sqrt{c} d e-5 a B e^2+12 A c d e-32 B c d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}+\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (\frac{a e^2}{(d+e x)^2}+c \left (\frac{d}{d+e x}-1\right )^2\right ) \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right )+\frac{\sqrt{c} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \left (-9 a A e^3+29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right )}{15 e^6 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right ) \sqrt{a+\frac{c (d+e x)^2 \left (\frac{d}{d+e x}-1\right )^2}{e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*Sqrt[a + c*x^2]*((2*B*c)/(3*e^4) - (2*(-(B*d) + A*e)*(c*d^2 + a*e^
2))/(5*e^4*(d + e*x)^3) + (2*(-17*B*c*d^2 + 12*A*c*d*e - 5*a*B*e^2))/(15*e^4*(d
+ e*x)^2) - (2*c*(-73*B*c*d^3 + 33*A*c*d^2*e - 61*a*B*d*e^2 + 21*a*A*e^3))/(15*e
^4*(c*d^2 + a*e^2)*(d + e*x))) - (8*c*(d + e*x)^(3/2)*(Sqrt[-d - (I*Sqrt[a]*e)/S
qrt[c]]*(32*B*c*d^3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*((a*e^2)/(d + e*x
)^2 + c*(-1 + d/(d + e*x))^2) + (Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(32*B*c*d^
3 - 12*A*c*d^2*e + 29*a*B*d*e^2 - 9*a*A*e^3)*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*Sqrt[a]*e)])/Sqrt[d + e*x] + (Sqrt[a]*e*(Sqrt[c]*
d + I*Sqrt[a]*e)*(-32*B*c*d^2 + (24*I)*Sqrt[a]*B*Sqrt[c]*d*e + 12*A*c*d*e - 5*a*
B*e^2 - (9*I)*Sqrt[a]*A*Sqrt[c]*e^2)*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))]*Ellipti
cF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqr
t[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(15*e^6*Sqrt[-d - (I*Sqrt[a]
*e)/Sqrt[c]]*(c*d^2 + a*e^2)*Sqrt[a + (c*(d + e*x)^2*(-1 + d/(d + e*x))^2)/e^2])

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Maple [B]  time = 0.067, size = 7383, normalized size = 13.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError