3.652 \(\int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=497 \[ \frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

[Out]

(4*Sqrt[d + e*x]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*
a*e^2)*x)*Sqrt[a + c*x^2])/(1155*c*e^3) + (2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28
*c*d*e*x)*(a + c*x^2)^(3/2))/(231*c*e) + (2*e*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(
11*c) + (32*Sqrt[-a]*d*(c*d^2 - 3*a*e^2)*(c*d^2 + 9*a*e^2)*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)])/(1155*Sqrt[c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]
*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(4*c^2*d^4 + 21
*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sq
rt[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)])/(1155*c^(3/2)*e^4*Sqrt[d + e*x]*Sqrt[a + c*x^2
])

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Rubi [A]  time = 1.54842, antiderivative size = 497, 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 \[ \frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[d + e*x]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*
a*e^2)*x)*Sqrt[a + c*x^2])/(1155*c*e^3) + (2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28
*c*d*e*x)*(a + c*x^2)^(3/2))/(231*c*e) + (2*e*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(
11*c) + (32*Sqrt[-a]*d*(c*d^2 - 3*a*e^2)*(c*d^2 + 9*a*e^2)*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)])/(1155*Sqrt[c]*e^4*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]
*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2 + a*e^2)*(4*c^2*d^4 + 21
*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sq
rt[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)])/(1155*c^(3/2)*e^4*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((e*x+d)**(3/2)*(c*x**2+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.28034, size = 695, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )+\frac{4 \left (-4 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-27 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+\sqrt{a} e (d+e x)^{3/2} \left (114 i a^{3/2} c d^2 e^3-15 i a^{5/2} e^5-108 a^2 \sqrt{c} d e^4+24 a c^{3/2} d^3 e^2+i \sqrt{a} c^2 d^4 e+4 c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (-6 a^{3/2} c d^2 e^3+27 a^{5/2} e^5-27 i a^2 \sqrt{c} d e^4+6 i a c^{3/2} d^3 e^2-\sqrt{a} c^2 d^4 e+i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{1155 c e^5 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(60*a^2*e^4 + a*c*e^2*(47*d^2 + 326*d*e*x + 19
5*e^2*x^2) + c^2*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^
4)) + (4*(-4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c^2*d^4 + 6*a*c*d^2*e^2 - 2
7*a^2*e^4)*(a + c*x^2) + 4*Sqrt[c]*d*(I*c^(5/2)*d^5 - Sqrt[a]*c^2*d^4*e + (6*I)*
a*c^(3/2)*d^3*e^2 - 6*a^(3/2)*c*d^2*e^3 - (27*I)*a^2*Sqrt[c]*d*e^4 + 27*a^(5/2)*
e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c]
- e*x)/(d + e*x))]*(d + e*x)^(3/2)*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[a]*e*(4*c^(5/2)*d^5 + I*Sqrt[a]*c^2*d^4*e + 24*a*c^(3/2)*d^3*e^2 + (114*I)*
a^(3/2)*c*d^2*e^3 - 108*a^2*Sqrt[c]*d*e^4 - (15*I)*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt
[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(
d + e*x)^(3/2)*EllipticF[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 - (I*Sqrt[a]
*e)/Sqrt[c]]*(d + e*x))))/(1155*c*e^5*Sqrt[a + c*x^2])

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Maple [B]  time = 0.065, size = 1969, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-16*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/
2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a
*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^4*d^7-60*a^3*c*d*e^6-47*a^2*c^2*
d^3*e^4-8*a*c^3*d^5*e^2-245*x^6*c^4*d*e^6-300*x^5*a*c^3*e^7-145*x^5*c^4*d^2*e^5+
x^4*c^4*d^3*e^4-255*x^3*a^2*c^2*e^7-2*x^3*c^4*d^4*e^3-8*x^2*c^4*d^5*e^2-60*x*a^3
*c*e^7-46*x^2*a*c^3*d^3*e^4-373*x*a^2*c^2*d^2*e^5-2*x*a*c^3*d^4*e^3-766*x^4*a*c^
3*d*e^6-360*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*
c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*Ellipti
cF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*
e+c*d))^(1/2))*a^2*c^2*d^3*e^4-60*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)
*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)
*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^3*e^7-518*x^3*a*c^3*d^2*e^5-
581*x^2*a^2*c^2*d*e^6-372*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))
^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/
((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d*e^6-112*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(
1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((
-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((
-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^5*e^2+432*(-(e*x+d)*c/((
-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c
*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/
2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d*e^6
+336*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))
^(1/2))*a^2*c^2*d^3*e^4+12*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)
^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d)
)^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)
/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^5*e^2+16*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))
^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-
((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*c^3*d^6*e+24*(-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))
^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(
-a*c)^(1/2)*a^2*c*d^2*e^5+100*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a
*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c
*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c
*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a*c^2*d^4*e^3-105*x^7*c^4*e^7)/c^2
/e^5/(c*e*x^3+c*d*x^2+a*e*x+a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*e*x^3 + c*d*x^2 + a*e*x + a*d)*sqrt(c*x^2 + a)*sqrt(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+a)**(3/2),x)

[Out]

Integral((a + c*x**2)**(3/2)*(d + e*x)**(3/2), x)

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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)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError