3.1671 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx\)

Optimal. Leaf size=178 \[ \frac{5 b^{3/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{11/4}}+\frac{5 d^2 \sqrt{a+b x}}{(c+d x)^{3/4} (b c-a d)^3}+\frac{3 d}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)} \]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/4)) + (3*d)/((b*c - a*d)^2*Sqrt[a
+ b*x]*(c + d*x)^(3/4)) + (5*d^2*Sqrt[a + b*x])/((b*c - a*d)^3*(c + d*x)^(3/4))
+ (5*b^(3/4)*d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c
+ d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/((b*c - a*d)^(11/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.275959, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 b^{3/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{11/4}}+\frac{5 d^2 \sqrt{a+b x}}{(c+d x)^{3/4} (b c-a d)^3}+\frac{3 d}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(7/4)),x]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/4)) + (3*d)/((b*c - a*d)^2*Sqrt[a
+ b*x]*(c + d*x)^(3/4)) + (5*d^2*Sqrt[a + b*x])/((b*c - a*d)^3*(c + d*x)^(3/4))
+ (5*b^(3/4)*d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c
+ d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/((b*c - a*d)^(11/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 42.2773, size = 231, normalized size = 1.3 \[ - \frac{5 b^{\frac{3}{4}} d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{2 \left (a d - b c\right )^{\frac{11}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{5 d^{2} \sqrt{a + b x}}{\left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )^{3}} + \frac{3 d}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{2}{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(7/4),x)

[Out]

-5*b**(3/4)*d*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)
/sqrt(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)/(2*(a*d - b*c)**(11
/4)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 5*d**2*sqrt(a + b*x)/((c + d*x)**(3/4)*(a
*d - b*c)**3) + 3*d/(sqrt(a + b*x)*(c + d*x)**(3/4)*(a*d - b*c)**2) + 2/(3*(a +
b*x)**(3/2)*(c + d*x)**(3/4)*(a*d - b*c))

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Mathematica [C]  time = 0.279332, size = 139, normalized size = 0.78 \[ \frac{-4 a^2 d^2-15 b d (a+b x) (c+d x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-a b d (13 c+21 d x)+b^2 \left (2 c^2-9 c d x-15 d^2 x^2\right )}{3 (a+b x)^{3/2} (c+d x)^{3/4} (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(7/4)),x]

[Out]

(-4*a^2*d^2 - a*b*d*(13*c + 21*d*x) + b^2*(2*c^2 - 9*c*d*x - 15*d^2*x^2) - 15*b*
d*(a + b*x)*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)^3*(a + b*x)^(3/2)*(c + d
*x)^(3/4))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/(d*x+c)^(7/4),x)

[Out]

int(1/(b*x+a)^(5/2)/(d*x+c)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(7/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(7/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(7/4)),x, algorithm="fricas")

[Out]

integral(1/((b^2*d*x^3 + a^2*c + (b^2*c + 2*a*b*d)*x^2 + (2*a*b*c + a^2*d)*x)*sq
rt(b*x + a)*(d*x + c)^(3/4)), 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(1/(b*x+a)**(5/2)/(d*x+c)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(7/4)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/2)*(d*x + c)^(7/4)), x)