3.1670 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{7/4}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{10 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt{a+b x} (b c-a d)^{7/4}}-\frac{10 d \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)} \]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4)) - (10*d*Sqrt[a + b*x])/(3*(b*c -
a*d)^2*(c + d*x)^(3/4)) - (10*b^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellipti
cF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*(b*c - a*d)^(7/4
)*Sqrt[a + b*x])

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Rubi [A]  time = 0.206575, antiderivative size = 146, 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 \[ -\frac{10 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt{a+b x} (b c-a d)^{7/4}}-\frac{10 d \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(3/2)*(c + d*x)^(7/4)),x]

[Out]

-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/4)) - (10*d*Sqrt[a + b*x])/(3*(b*c -
a*d)^2*(c + d*x)^(3/4)) - (10*b^(3/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Ellipti
cF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(3*(b*c - a*d)^(7/4
)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 30.535, size = 199, normalized size = 1.36 \[ - \frac{5 b^{\frac{3}{4}} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 \left (a d - b c\right )^{\frac{7}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{10 d \sqrt{a + b x}}{3 \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{2}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(7/4),x)

[Out]

-5*b**(3/4)*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/s
qrt(a*d - b*c) + 1)**2))*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_f(
2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(3*(a*d - b*c)**(7/4)
*sqrt(a - b*c/d + b*(c + d*x)/d)) - 10*d*sqrt(a + b*x)/(3*(c + d*x)**(3/4)*(a*d
- b*c)**2) + 2/(sqrt(a + b*x)*(c + d*x)**(3/4)*(a*d - b*c))

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Mathematica [C]  time = 0.21511, size = 102, normalized size = 0.7 \[ -\frac{2 \left (5 b (c+d x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+2 a d+3 b c+5 b d x\right )}{3 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(7/4)),x]

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(3/2)/(d*x+c)^(7/4),x)

[Out]

int(1/(b*x+a)^(3/2)/(d*x+c)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(7/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(7/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(7/4)),x, algorithm="fricas")

[Out]

integral(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*sqrt(b*x + a)*(d*x + c)^(3/4)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(3/2)/(d*x+c)**(7/4),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(7/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(7/4)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(7/4)), x)