3.1497 \(\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]

[Out]

(-2*Sqrt[c + d*x])/((b*c - a*d)*Sqrt[a + b*x])

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Rubi [A]  time = 0.0241024, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c + d*x])/((b*c - a*d)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 3.78672, size = 24, normalized size = 0.8 \[ \frac{2 \sqrt{c + d x}}{\sqrt{a + b x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c + d*x)/(sqrt(a + b*x)*(a*d - b*c))

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Mathematica [A]  time = 0.0327016, size = 30, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x])/((-(b*c) + a*d)*Sqrt[a + b*x])

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Maple [A]  time = 0.007, size = 27, normalized size = 0.9 \[ 2\,{\frac{\sqrt{dx+c}}{\sqrt{bx+a} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b*x+a)^(1/2)*(d*x+c)^(1/2)/(a*d-b*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223804, size = 57, normalized size = 1.9 \[ -\frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c - a^2*d + (b^2*c - a*b*d)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229165, size = 89, normalized size = 2.97 \[ -\frac{4 \, \sqrt{b d} b}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(b*d)*b/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2)*abs(b))