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

Optimal. Leaf size=214 \[ \frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} \sqrt{d}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b \sqrt{a+b x} (b c-a d)} \]

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4) + (5*(b*c - 7*
a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^3) + ((b*c - 7*a*d)*Sqrt[a + b*x]*(c +
d*x)^(5/2))/(3*b^2*(b*c - a*d)) + (2*a*(c + d*x)^(7/2))/(b*(b*c - a*d)*Sqrt[a +
b*x]) + (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(9/2)*Sqrt[d])

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Rubi [A]  time = 0.303986, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} \sqrt{d}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{b \sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b*c - 7*a*d)*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4) + (5*(b*c - 7*
a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^3) + ((b*c - 7*a*d)*Sqrt[a + b*x]*(c +
d*x)^(5/2))/(3*b^2*(b*c - a*d)) + (2*a*(c + d*x)^(7/2))/(b*(b*c - a*d)*Sqrt[a +
b*x]) + (5*(b*c - 7*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(9/2)*Sqrt[d])

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Rubi in Sympy [A]  time = 36.2835, size = 196, normalized size = 0.92 \[ - \frac{2 a \left (c + d x\right )^{\frac{7}{2}}}{b \sqrt{a + b x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d - b c\right )}{3 b^{2} \left (a d - b c\right )} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (7 a d - b c\right )}{12 b^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (7 a d - b c\right )}{8 b^{4}} - \frac{5 \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{9}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(c + d*x)**(7/2)/(b*sqrt(a + b*x)*(a*d - b*c)) + sqrt(a + b*x)*(c + d*x)**(
5/2)*(7*a*d - b*c)/(3*b**2*(a*d - b*c)) - 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(7*a*
d - b*c)/(12*b**3) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)*(7*a*d - b*c)/(8*
b**4) - 5*(a*d - b*c)**2*(7*a*d - b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt
(c + d*x)))/(8*b**(9/2)*sqrt(d))

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Mathematica [A]  time = 0.203599, size = 173, normalized size = 0.81 \[ \frac{\sqrt{c+d x} \left (105 a^3 d^2+5 a^2 b d (7 d x-38 c)+a b^2 \left (81 c^2-68 c d x-14 d^2 x^2\right )+b^3 x \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{24 b^4 \sqrt{a+b x}}+\frac{5 (b c-7 a d) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{9/2} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(105*a^3*d^2 + 5*a^2*b*d*(-38*c + 7*d*x) + a*b^2*(81*c^2 - 68*c*d
*x - 14*d^2*x^2) + b^3*x*(33*c^2 + 26*c*d*x + 8*d^2*x^2)))/(24*b^4*Sqrt[a + b*x]
) + (5*(b*c - 7*a*d)*(b*c - a*d)^2*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*S
qrt[a + b*x]*Sqrt[c + d*x]])/(16*b^(9/2)*Sqrt[d])

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Maple [B]  time = 0.037, size = 689, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(d*x+c)^(1/2)*(-16*x^3*b^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(
1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b
*d^3-225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*x*a^2*b^2*c*d^2+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*x*a*b^3*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*
(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^4*c^3+28*x^2*a*b^2*d^2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)-52*x^2*b^3*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(
2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^3-225*
ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*
b*c*d^2+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)
^(1/2))*a^2*b^2*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*a*b^3*c^3-70*x*a^2*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+136*x*a*b^2*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-66*x*b^3*c^2*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)-210*a^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+380*a^2*b
*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-162*a*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.580334, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, b^{3} d^{2} x^{3} + 81 \, a b^{2} c^{2} - 190 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (13 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} +{\left (33 \, b^{3} c^{2} - 68 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b d}}, \frac{2 \,{\left (8 \, b^{3} d^{2} x^{3} + 81 \, a b^{2} c^{2} - 190 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (13 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} +{\left (33 \, b^{3} c^{2} - 68 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^3*d^2*x^3 + 81*a*b^2*c^2 - 190*a^2*b*c*d + 105*a^3*d^2 + 2*(13*b^3
*c*d - 7*a*b^2*d^2)*x^2 + (33*b^3*c^2 - 68*a*b^2*c*d + 35*a^2*b*d^2)*x)*sqrt(b*d
)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2
- 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 - 7*a^3*b*d^3)*x)*log
(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x
^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/((b^5*
x + a*b^4)*sqrt(b*d)), 1/48*(2*(8*b^3*d^2*x^3 + 81*a*b^2*c^2 - 190*a^2*b*c*d + 1
05*a^3*d^2 + 2*(13*b^3*c*d - 7*a*b^2*d^2)*x^2 + (33*b^3*c^2 - 68*a*b^2*c*d + 35*
a^2*b*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(a*b^3*c^3 - 9*a^2*b^2
*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*a^2*b^2*c*d^
2 - 7*a^3*b*d^3)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*s
qrt(d*x + c)*b*d)))/((b^5*x + a*b^4)*sqrt(-b*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.62004, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x