3.758 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{15 \sqrt{d} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2}}+\frac{15 d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b^3}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2}}{2 b^2}-\frac{2 (c+d x)^{5/2}}{b \sqrt{a+b x}} \]

[Out]

(15*d*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^3) + (5*d*Sqrt[a + b*x]*(c +
d*x)^(3/2))/(2*b^2) - (2*(c + d*x)^(5/2))/(b*Sqrt[a + b*x]) + (15*Sqrt[d]*(b*c
- a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2))

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Rubi [A]  time = 0.168612, antiderivative size = 138, 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 \[ \frac{15 \sqrt{d} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2}}+\frac{15 d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b^3}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2}}{2 b^2}-\frac{2 (c+d x)^{5/2}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(15*d*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^3) + (5*d*Sqrt[a + b*x]*(c +
d*x)^(3/2))/(2*b^2) - (2*(c + d*x)^(5/2))/(b*Sqrt[a + b*x]) + (15*Sqrt[d]*(b*c
- a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 21.1569, size = 128, normalized size = 0.93 \[ - \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{b \sqrt{a + b x}} + \frac{5 d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 b^{2}} - \frac{15 d \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 b^{3}} + \frac{15 \sqrt{d} \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c + d*x)**(5/2)/(b*sqrt(a + b*x)) + 5*d*sqrt(a + b*x)*(c + d*x)**(3/2)/(2*b*
*2) - 15*d*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)/(4*b**3) + 15*sqrt(d)*(a*d -
b*c)**2*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(4*b**(7/2))

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Mathematica [A]  time = 0.162094, size = 138, normalized size = 1. \[ \frac{15 \sqrt{d} (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{7/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 (b c-a d)^2}{b^3 (a+b x)}+\frac{d (9 b c-7 a d)}{4 b^3}+\frac{d^2 x}{2 b^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((d*(9*b*c - 7*a*d))/(4*b^3) + (d^2*x)/(2*b^2) - (2*
(b*c - a*d)^2)/(b^3*(a + b*x))) + (15*Sqrt[d]*(b*c - a*d)^2*Log[b*c + a*d + 2*b*
d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*b^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((d*x + c)^(5/2)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.399985, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (2 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 25 \, a b c d - 15 \, a^{2} d^{2} +{\left (9 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{15 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 25 \, a b c d - 15 \, a^{2} d^{2} +{\left (9 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^
2)*x)*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x
+ b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*
x) + 4*(2*b^2*d^2*x^2 - 8*b^2*c^2 + 25*a*b*c*d - 15*a^2*d^2 + (9*b^2*c*d - 5*a*b
*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*x + a*b^3), 1/8*(15*(a*b^2*c^2 - 2*a^
2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x)*sqrt(-d/b)*arctan(1/2
*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) + 2*(2*b^2*d^
2*x^2 - 8*b^2*c^2 + 25*a*b*c*d - 15*a^2*d^2 + (9*b^2*c*d - 5*a*b*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(b^4*x + a*b^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.61658, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x