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

Optimal. Leaf size=248 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+10 a c-\frac{35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{5/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b d^2} \]

[Out]

(2*c^2*(a + b*x)^(5/2))/(d^2*(b*c - a*d)*Sqrt[c + d*x]) + ((35*b^2*c^2 - 10*a*b*
c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b*d^4) + ((10*a*c - (35*b*c^2)/d
+ (a^2*d)/b)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^2*(b*c - a*d)) + ((a + b*x)^(5
/2)*Sqrt[c + d*x])/(3*b*d^2) - ((b*c - a*d)*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*
ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(9/2))

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Rubi [A]  time = 0.596707, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+10 a c-\frac{35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{5/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*(a + b*x)^(5/2))/(d^2*(b*c - a*d)*Sqrt[c + d*x]) + ((35*b^2*c^2 - 10*a*b*
c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b*d^4) + ((10*a*c - (35*b*c^2)/d
+ (a^2*d)/b)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^2*(b*c - a*d)) + ((a + b*x)^(5
/2)*Sqrt[c + d*x])/(3*b*d^2) - ((b*c - a*d)*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*
ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(3/2)*d^(9/2))

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Rubi in Sympy [A]  time = 44.1898, size = 228, normalized size = 0.92 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{5}{2}}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 b d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right )}{12 b d^{3} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right )}{8 b d^{4}} - \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c**2*(a + b*x)**(5/2)/(d**2*sqrt(c + d*x)*(a*d - b*c)) + (a + b*x)**(5/2)*sqr
t(c + d*x)/(3*b*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a**2*d**2 + 10*a*b*c*d -
35*b**2*c**2)/(12*b*d**3*(a*d - b*c)) - sqrt(a + b*x)*sqrt(c + d*x)*(a**2*d**2
+ 10*a*b*c*d - 35*b**2*c**2)/(8*b*d**4) - (a*d - b*c)*(a**2*d**2 + 10*a*b*c*d -
35*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(8*b**(3/2)*d
**(9/2))

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Mathematica [A]  time = 0.19361, size = 189, normalized size = 0.76 \[ \frac{\sqrt{a+b x} \left (3 a^2 d^2 (c+d x)+2 a b d \left (-50 c^2-19 c d x+7 d^2 x^2\right )+b^2 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{24 b d^4 \sqrt{c+d x}}-\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{3/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(3*a^2*d^2*(c + d*x) + 2*a*b*d*(-50*c^2 - 19*c*d*x + 7*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*b*d^4*Sqrt[c + d*x]
) - ((b*c - a*d)*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2
*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*b^(3/2)*d^(9/2))

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Maple [B]  time = 0.038, size = 692, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(3/2)/(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)+3*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
+27*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^3-135*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^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-28*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)+3*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+27*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^2-135*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*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^4-6*x*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+76*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)-6*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+200*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/(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)^(3/2)*x^2/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.597564, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \,{\left (b d^{5} x + b c d^{4}\right )} \sqrt{b d}}, \frac{2 \,{\left (8 \, b^{2} d^{3} x^{3} + 105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \,{\left (b d^{5} x + b c d^{4}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^2*d^3*x^3 + 105*b^2*c^3 - 100*a*b*c^2*d + 3*a^2*c*d^2 - 14*(b^2*c*
d^2 - a*b*d^3)*x^2 + (35*b^2*c^2*d - 38*a*b*c*d^2 + 3*a^2*d^3)*x)*sqrt(b*d)*sqrt
(b*x + a)*sqrt(d*x + c) + 3*(35*b^3*c^4 - 45*a*b^2*c^3*d + 9*a^2*b*c^2*d^2 + a^3
*c*d^3 + (35*b^3*c^3*d - 45*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)))/((b*d^5*x +
b*c*d^4)*sqrt(b*d)), 1/48*(2*(8*b^2*d^3*x^3 + 105*b^2*c^3 - 100*a*b*c^2*d + 3*a
^2*c*d^2 - 14*(b^2*c*d^2 - a*b*d^3)*x^2 + (35*b^2*c^2*d - 38*a*b*c*d^2 + 3*a^2*d
^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(35*b^3*c^4 - 45*a*b^2*c^3*d +
9*a^2*b*c^2*d^2 + a^3*c*d^3 + (35*b^3*c^3*d - 45*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)*sqrt(d*
x + c)*b*d)))/((b*d^5*x + b*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**2*(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258565, size = 428, normalized size = 1.73 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \, b^{3} c d^{5} + 5 \, a b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{35 \, b^{4} c^{2} d^{4} - 10 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{3 \,{\left (35 \, b^{5} c^{3} d^{3} - 45 \, a b^{4} c^{2} d^{4} + 9 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left (35 \, b^{2} c^{2} - 10 \, a b c d - a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{61440 \, \sqrt{b d} b^{7} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/184320*((2*(4*(b*x + a)*b^2*d^6/(b^10*c*d^8 - a*b^9*d^9) - (7*b^3*c*d^5 + 5*a*
b^2*d^6)/(b^10*c*d^8 - a*b^9*d^9))*(b*x + a) + (35*b^4*c^2*d^4 - 10*a*b^3*c*d^5
- a^2*b^2*d^6)/(b^10*c*d^8 - a*b^9*d^9))*(b*x + a) + 3*(35*b^5*c^3*d^3 - 45*a*b^
4*c^2*d^4 + 9*a^2*b^3*c*d^5 + a^3*b^2*d^6)/(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) + 1/61440*(35*b^2*c^2 - 10*a*b*c*d - a^2*
d^2)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sq
rt(b*d)*b^7*d^5)