3.1570 \(\int \frac{1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=842 \[ \frac{10 \sqrt{a+b x} d^2}{9 b^{2/3} (b c-a d)^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{5 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) d}{3\ 3^{3/4} b^{2/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{10 \sqrt{2} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) d}{9 \sqrt [4]{3} b^{2/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{10 (c+d x)^{2/3} d}{9 (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{2/3}}{3 (b c-a d) (a+b x)^{3/2}} \]

[Out]

(-2*(c + d*x)^(2/3))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (10*d*(c + d*x)^(2/3))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) + (10*d^2*Sqrt[a + b*x])/(9*b^(2/3)*(b*c - a*d)^2*
((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))) - (5*Sqrt[2 + Sqrt[
3]]*d*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^
(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3]
)*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3]
)*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3)
- b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*b^(2/3)*(b*c - a*d)^(5/
3)*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x
)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)]) + (10
*Sqrt[2]*d*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3)
+ b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sq
rt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sq
rt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(
1/3) - b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*b^(2/3)*(b*c - a*d
)^(5/3)*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c
+ d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 1.79538, antiderivative size = 842, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{10 \sqrt{a+b x} d^2}{9 b^{2/3} (b c-a d)^2 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{5 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) d}{3\ 3^{3/4} b^{2/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{10 \sqrt{2} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right ) d}{9 \sqrt [4]{3} b^{2/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{10 (c+d x)^{2/3} d}{9 (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{2/3}}{3 (b c-a d) (a+b x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(1/3)),x]

[Out]

(-2*(c + d*x)^(2/3))/(3*(b*c - a*d)*(a + b*x)^(3/2)) + (10*d*(c + d*x)^(2/3))/(9
*(b*c - a*d)^2*Sqrt[a + b*x]) + (10*d^2*Sqrt[a + b*x])/(9*b^(2/3)*(b*c - a*d)^2*
((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))) - (5*Sqrt[2 + Sqrt[
3]]*d*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^
(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3]
)*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3]
)*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3)
- b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*b^(2/3)*(b*c - a*d)^(5/
3)*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x
)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)]) + (10
*Sqrt[2]*d*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3)
+ b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sq
rt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sq
rt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(
1/3) - b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*b^(2/3)*(b*c - a*d
)^(5/3)*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c
+ d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 107.233, size = 729, normalized size = 0.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/3),x)

[Out]

10*d*(c + d*x)**(2/3)/(9*sqrt(a + b*x)*(a*d - b*c)**2) + 2*(c + d*x)**(2/3)/(3*(
a + b*x)**(3/2)*(a*d - b*c)) - 10*d**2*sqrt(a - b*c/d + b*(c + d*x)/d)/(9*b**(2/
3)*(a*d - b*c)**2*(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))
) + 5*3**(1/4)*d*sqrt((b**(2/3)*(c + d*x)**(2/3) - b**(1/3)*(c + d*x)**(1/3)*(a*
d - b*c)**(1/3) + (a*d - b*c)**(2/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))
*(a*d - b*c)**(1/3))**2)*sqrt(-sqrt(3) + 2)*(b**(1/3)*(c + d*x)**(1/3) + (a*d -
b*c)**(1/3))*elliptic_e(asin((b**(1/3)*(c + d*x)**(1/3) - (-1 + sqrt(3))*(a*d -
b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))), -7
- 4*sqrt(3))/(9*b**(2/3)*sqrt((a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (
a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3)
)**2)*(a*d - b*c)**(5/3)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 10*sqrt(2)*3**(3/4)*
d*sqrt((b**(2/3)*(c + d*x)**(2/3) - b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(1/3)
+ (a*d - b*c)**(2/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(
1/3))**2)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_f(asin((b**(
1/3)*(c + d*x)**(1/3) - (-1 + sqrt(3))*(a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**
(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))), -7 - 4*sqrt(3))/(27*b**(2/3)*sqrt((a
*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(c +
d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*(a*d - b*c)**(5/3)*sqrt(a -
b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.248248, size = 105, normalized size = 0.12 \[ \frac{(c+d x)^{2/3} \left (4 (8 a d-3 b c+5 b d x)-5 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{18 (a+b x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(1/3)),x]

[Out]

((c + d*x)^(2/3)*(4*(-3*b*c + 8*a*d + 5*b*d*x) - 5*d*(a + b*x)*Sqrt[(d*(a + b*x)
)/(-(b*c) + a*d)]*Hypergeometric2F1[1/2, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)]))/
(18*(b*c - a*d)^2*(a + b*x)^(3/2))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(1/3),x)

[Out]

int(1/(b*x+a)^(5/2)/(d*x+c)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*(d*x + c)^(1/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/3)), x)