3.1565 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=457 \[ -\frac{28 \sqrt{2-\sqrt{3}} d^2 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{135 \sqrt [4]{3} b^{4/3} \sqrt{a+b x} (b c-a d)^2 \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{28 d^2 \sqrt [3]{c+d x}}{135 b \sqrt{a+b x} (b c-a d)^2}-\frac{4 d \sqrt [3]{c+d x}}{45 b (a+b x)^{3/2} (b c-a d)}-\frac{2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}} \]

[Out]

(-2*(c + d*x)^(1/3))/(5*b*(a + b*x)^(5/2)) - (4*d*(c + d*x)^(1/3))/(45*b*(b*c -
a*d)*(a + b*x)^(3/2)) + (28*d^2*(c + d*x)^(1/3))/(135*b*(b*c - a*d)^2*Sqrt[a + b
*x]) - (28*Sqrt[2 - Sqrt[3]]*d^2*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*S
qrt[((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]*Ell
ipticF[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]])/(135*3^
(1/4)*b^(4/3)*(b*c - a*d)^2*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 = 0.768938, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{28 \sqrt{2-\sqrt{3}} d^2 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{135 \sqrt [4]{3} b^{4/3} \sqrt{a+b x} (b c-a d)^2 \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{28 d^2 \sqrt [3]{c+d x}}{135 b \sqrt{a+b x} (b c-a d)^2}-\frac{4 d \sqrt [3]{c+d x}}{45 b (a+b x)^{3/2} (b c-a d)}-\frac{2 \sqrt [3]{c+d x}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/3)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(1/3))/(5*b*(a + b*x)^(5/2)) - (4*d*(c + d*x)^(1/3))/(45*b*(b*c -
a*d)*(a + b*x)^(3/2)) + (28*d^2*(c + d*x)^(1/3))/(135*b*(b*c - a*d)^2*Sqrt[a + b
*x]) - (28*Sqrt[2 - Sqrt[3]]*d^2*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*S
qrt[((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]*Ell
ipticF[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]])/(135*3^
(1/4)*b^(4/3)*(b*c - a*d)^2*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 = 56.2976, size = 389, normalized size = 0.85 \[ \frac{28 d^{2} \sqrt [3]{c + d x}}{135 b \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{4 d \sqrt [3]{c + d x}}{45 b \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \sqrt [3]{c + d x}}{5 b \left (a + b x\right )^{\frac{5}{2}}} + \frac{28 \cdot 3^{\frac{3}{4}} d^{2} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{405 b^{\frac{4}{3}} \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \left (a d - b c\right )^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/3)/(b*x+a)**(7/2),x)

[Out]

28*d**2*(c + d*x)**(1/3)/(135*b*sqrt(a + b*x)*(a*d - b*c)**2) + 4*d*(c + d*x)**(
1/3)/(45*b*(a + b*x)**(3/2)*(a*d - b*c)) - 2*(c + d*x)**(1/3)/(5*b*(a + b*x)**(5
/2)) + 28*3**(3/4)*d**2*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 + s
qrt(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_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))/(405*b**(4/3)*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)**2*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.341816, size = 140, normalized size = 0.31 \[ \frac{2 \sqrt [3]{c+d x} \left (-7 a^2 d^2+7 d^2 (a+b x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+2 a b d (24 c+17 d x)+b^2 \left (-27 c^2-6 c d x+14 d^2 x^2\right )\right )}{135 b (a+b x)^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/3)/(a + b*x)^(7/2),x]

[Out]

(2*(c + d*x)^(1/3)*(-7*a^2*d^2 + 2*a*b*d*(24*c + 17*d*x) + b^2*(-27*c^2 - 6*c*d*
x + 14*d^2*x^2) + 7*d^2*(a + b*x)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeom
etric2F1[1/3, 1/2, 4/3, (b*(c + d*x))/(b*c - a*d)]))/(135*b*(b*c - a*d)^2*(a + b
*x)^(5/2))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/3)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(1/3)/(b*x+a)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/3)/(b*x+a)**(7/2),x)

[Out]

Integral((c + d*x)**(1/3)/(a + b*x)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(7/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(7/2), x)