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

Optimal. Leaf size=218 \[ -\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (7 b c-a d)}{12 d^3}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{d \sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.338011, antiderivative size = 218, 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 \[ -\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) (7 b c-a d)}{8 d^4}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (7 b c-a d)}{12 d^3}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 b c-a d)}{3 d^2 (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{d \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.2334, size = 196, normalized size = 0.9 \[ \frac{2 c \left (a + b x\right )^{\frac{7}{2}}}{d \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - 7 b c\right )}{3 d^{2} \left (a d - b c\right )} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - 7 b c\right )}{12 d^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - 7 b c\right ) \left (a d - b c\right )}{8 d^{4}} + \frac{5 \left (a d - 7 b c\right ) \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 \sqrt{b} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.220244, size = 176, normalized size = 0.81 \[ \frac{\sqrt{a+b x} \left (3 a^2 d^2 (27 c+11 d x)+2 a b d \left (-95 c^2-34 c d x+13 d^2 x^2\right )+b^2 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{24 d^4 \sqrt{c+d x}}+\frac{5 (a d-7 b c) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 \sqrt{b} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(3*a^2*d^2*(27*c + 11*d*x) + 2*a*b*d*(-95*c^2 - 34*c*d*x + 13*d^2
*x^2) + b^2*(105*c^3 + 35*c^2*d*x - 14*c*d^2*x^2 + 8*d^3*x^3)))/(24*d^4*Sqrt[c +
d*x]) + (5*(b*c - a*d)^2*(-7*b*c + a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqr
t[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*Sqrt[b]*d^(9/2))

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

[Out]

1/48*(b*x+a)^(1/2)*(16*x^3*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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*a^3*d^4-
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^2*b*c*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*b^2*c^2*d^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*b^3*c^3*d+52*x^2*a*b*d^3*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)-28*x^2*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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^3*c*d^3-135*l
n(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
*c^2*d^2+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*b^2*c^3*d-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))*b^3*c^4+66*x*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-13
6*x*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+70*x*b^2*c^2*d*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+162*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-380*a*b*c
^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+210*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(d*x+c)^(1/2)/d^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((b*x + a)^(5/2)*x/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.664684, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 190 \, a b c^{2} d + 81 \, a^{2} c d^{2} - 2 \,{\left (7 \, b^{2} c d^{2} - 13 \, a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 68 \, a b c d^{2} + 33 \, a^{2} d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \,{\left (d^{5} x + c d^{4}\right )} \sqrt{b d}}, \frac{2 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 190 \, a b c^{2} d + 81 \, a^{2} c d^{2} - 2 \,{\left (7 \, b^{2} c d^{2} - 13 \, a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 68 \, a b c d^{2} + 33 \, a^{2} d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (7 \, b^{3} c^{4} - 15 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (7 \, b^{3} c^{3} d - 15 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \,{\left (d^{5} x + c d^{4}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250129, size = 452, normalized size = 2.07 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b d^{6}{\left | b \right |}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \, b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{5 \,{\left (7 \, b^{3} c^{2} d^{4}{\left | b \right |} - 8 \, a b^{2} c d^{5}{\left | b \right |} + a^{2} b d^{6}{\left | b \right |}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (7 \, b^{4} c^{3} d^{3}{\left | b \right |} - 15 \, a b^{3} c^{2} d^{4}{\left | b \right |} + 9 \, a^{2} b^{2} c d^{5}{\left | b \right |} - a^{3} b d^{6}{\left | b \right |}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left (7 \, b^{2} c^{2}{\left | b \right |} - 8 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{12288 \, \sqrt{b d} b^{8} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/184320*((2*(4*(b*x + a)*b*d^6*abs(b)/(b^10*c*d^8 - a*b^9*d^9) - (7*b^2*c*d^5*a
bs(b) - a*b*d^6*abs(b))/(b^10*c*d^8 - a*b^9*d^9))*(b*x + a) + 5*(7*b^3*c^2*d^4*a
bs(b) - 8*a*b^2*c*d^5*abs(b) + a^2*b*d^6*abs(b))/(b^10*c*d^8 - a*b^9*d^9))*(b*x
+ a) + 15*(7*b^4*c^3*d^3*abs(b) - 15*a*b^3*c^2*d^4*abs(b) + 9*a^2*b^2*c*d^5*abs(
b) - a^3*b*d^6*abs(b))/(b^10*c*d^8 - a*b^9*d^9))*sqrt(b*x + a)/sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d) + 1/12288*(7*b^2*c^2*abs(b) - 8*a*b*c*d*abs(b) + a^2*d^2*abs(
b))*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqr
t(b*d)*b^8*d^5)