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

Optimal. Leaf size=222 \[ \frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{c+d x} (7 b c-3 a d)}{4 d^4}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac{2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

(-2*c*(a + b*x)^(7/2))/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - (2*(7*b*c - 3*a*d)*(a
+ b*x)^(5/2))/(3*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (5*b*(7*b*c - 3*a*d)*Sqrt[a +
b*x]*Sqrt[c + d*x])/(4*d^4) + (5*b*(7*b*c - 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x
])/(6*d^3*(b*c - a*d)) + (5*Sqrt[b]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]
*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(9/2))

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Rubi [A]  time = 0.321757, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{c+d x} (7 b c-3 a d)}{4 d^4}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac{2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(a + b*x)^(7/2))/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - (2*(7*b*c - 3*a*d)*(a
+ b*x)^(5/2))/(3*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (5*b*(7*b*c - 3*a*d)*Sqrt[a +
b*x]*Sqrt[c + d*x])/(4*d^4) + (5*b*(7*b*c - 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x
])/(6*d^3*(b*c - a*d)) + (5*Sqrt[b]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]
*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(9/2))

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Rubi in Sympy [A]  time = 35.0257, size = 207, normalized size = 0.93 \[ \frac{5 \sqrt{b} \left (a d - b c\right ) \left (3 a d - 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 d^{\frac{9}{2}}} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{6 d^{3} \left (a d - b c\right )} + \frac{5 b \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{4 d^{4}} + \frac{2 c \left (a + b x\right )^{\frac{7}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (3 a d - 7 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(b)*(a*d - b*c)*(3*a*d - 7*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(
c + d*x)))/(4*d**(9/2)) + 5*b*(a + b*x)**(3/2)*sqrt(c + d*x)*(3*a*d - 7*b*c)/(6*
d**3*(a*d - b*c)) + 5*b*sqrt(a + b*x)*sqrt(c + d*x)*(3*a*d - 7*b*c)/(4*d**4) + 2
*c*(a + b*x)**(7/2)/(3*d*(c + d*x)**(3/2)*(a*d - b*c)) - 2*(a + b*x)**(5/2)*(3*a
*d - 7*b*c)/(3*d**2*sqrt(c + d*x)*(a*d - b*c))

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Mathematica [A]  time = 0.263988, size = 175, normalized size = 0.79 \[ \frac{\sqrt{a+b x} \left (-8 a^2 d^2 (2 c+3 d x)+a b d \left (115 c^2+158 c d x+27 d^2 x^2\right )+b^2 \left (-\left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )\right )}{12 d^4 (c+d x)^{3/2}}+\frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(-8*a^2*d^2*(2*c + 3*d*x) + a*b*d*(115*c^2 + 158*c*d*x + 27*d^2*x
^2) - b^2*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3)))/(12*d^4*(c + d*x)
^(3/2)) + (5*Sqrt[b]*(7*b*c - 3*a*d)*(b*c - a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqr
t[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*d^(9/2))

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Maple [B]  time = 0.034, size = 750, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(b*x+a)^(1/2)*(45*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^2*a^2*b*d^4-150*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^2*a*b^2*c*d^3+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^2*b^3*c^2*d^2+12*x^3*b^2*d^3*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+90*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^3-300*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+210*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+54*
x^2*a*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-42*x^2*b^2*c*d^2*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+45*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*c^2*d^2-150*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^4-48*x*a^2*d^3*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)+316*x*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-280*x
*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-32*a^2*c*d^2*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+230*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)^(
3/2)/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.701383, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \,{\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, \frac{15 \,{\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) + 2 \,{\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \,{\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*(7*b^2*c^4 - 10*a*b*c^3*d + 3*a^2*c^2*d^2 + (7*b^2*c^2*d^2 - 10*a*b*c*
d^3 + 3*a^2*d^4)*x^2 + 2*(7*b^2*c^3*d - 10*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(b/
d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*
d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(6*b^2
*d^3*x^3 - 105*b^2*c^3 + 115*a*b*c^2*d - 16*a^2*c*d^2 - 3*(7*b^2*c*d^2 - 9*a*b*d
^3)*x^2 - 2*(70*b^2*c^2*d - 79*a*b*c*d^2 + 12*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x
+ c))/(d^6*x^2 + 2*c*d^5*x + c^2*d^4), 1/24*(15*(7*b^2*c^4 - 10*a*b*c^3*d + 3*a
^2*c^2*d^2 + (7*b^2*c^2*d^2 - 10*a*b*c*d^3 + 3*a^2*d^4)*x^2 + 2*(7*b^2*c^3*d - 1
0*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqr
t(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + 2*(6*b^2*d^3*x^3 - 105*b^2*c^3 + 115*a
*b*c^2*d - 16*a^2*c*d^2 - 3*(7*b^2*c*d^2 - 9*a*b*d^3)*x^2 - 2*(70*b^2*c^2*d - 79
*a*b*c*d^2 + 12*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^6*x^2 + 2*c*d^5*x +
c^2*d^4)]

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261446, size = 545, normalized size = 2.45 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{5} c d^{6}{\left | b \right |} - a b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{4} c d^{7} - a b^{3} d^{8}} - \frac{7 \, b^{6} c^{2} d^{5}{\left | b \right |} - 10 \, a b^{5} c d^{6}{\left | b \right |} + 3 \, a^{2} b^{4} d^{7}{\left | b \right |}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} - \frac{20 \,{\left (7 \, b^{7} c^{3} d^{4}{\left | b \right |} - 17 \, a b^{6} c^{2} d^{5}{\left | b \right |} + 13 \, a^{2} b^{5} c d^{6}{\left | b \right |} - 3 \, a^{3} b^{4} d^{7}{\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, b^{8} c^{4} d^{3}{\left | b \right |} - 24 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 30 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} - 16 \, a^{3} b^{5} c d^{6}{\left | b \right |} + 3 \, a^{4} b^{4} d^{7}{\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (7 \, b^{2} c^{2}{\left | b \right |} - 10 \, a b c d{\left | b \right |} + 3 \, a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((3*(b*x + a)*(2*(b^5*c*d^6*abs(b) - a*b^4*d^7*abs(b))*(b*x + a)/(b^4*c*d^7
- a*b^3*d^8) - (7*b^6*c^2*d^5*abs(b) - 10*a*b^5*c*d^6*abs(b) + 3*a^2*b^4*d^7*ab
s(b))/(b^4*c*d^7 - a*b^3*d^8)) - 20*(7*b^7*c^3*d^4*abs(b) - 17*a*b^6*c^2*d^5*abs
(b) + 13*a^2*b^5*c*d^6*abs(b) - 3*a^3*b^4*d^7*abs(b))/(b^4*c*d^7 - a*b^3*d^8))*(
b*x + a) - 15*(7*b^8*c^4*d^3*abs(b) - 24*a*b^7*c^3*d^4*abs(b) + 30*a^2*b^6*c^2*d
^5*abs(b) - 16*a^3*b^5*c*d^6*abs(b) + 3*a^4*b^4*d^7*abs(b))/(b^4*c*d^7 - a*b^3*d
^8))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/4*(7*b^2*c^2*abs(b)
- 10*a*b*c*d*abs(b) + 3*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)))/(sqrt(b*d)*d^4)