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

Optimal. Leaf size=309 \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{64 b d^5}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{96 b d^4}+\frac{5 (b c-a d)^2 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{11/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+14 a c-\frac{63 b c^2}{d}\right )}{24 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{7/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d^2} \]

[Out]

(2*c^2*(a + b*x)^(7/2))/(d^2*(b*c - a*d)*Sqrt[c + d*x]) - (5*(b*c - a*d)*(63*b^2
*c^2 - 14*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^5) + (5*(63*b^
2*c^2 - 14*a*b*c*d - a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^4) + ((14*a
*c - (63*b*c^2)/d + (a^2*d)/b)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*d^2*(b*c - a*d
)) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b*d^2) + (5*(b*c - a*d)^2*(63*b^2*c^2 -
14*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/
(64*b^(3/2)*d^(11/2))

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Rubi [A]  time = 0.751876, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{64 b d^5}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right )}{96 b d^4}+\frac{5 (b c-a d)^2 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{11/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (\frac{a^2 d}{b}+14 a c-\frac{63 b c^2}{d}\right )}{24 d^2 (b c-a d)}+\frac{2 c^2 (a+b x)^{7/2}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*(a + b*x)^(7/2))/(d^2*(b*c - a*d)*Sqrt[c + d*x]) - (5*(b*c - a*d)*(63*b^2
*c^2 - 14*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^5) + (5*(63*b^
2*c^2 - 14*a*b*c*d - a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^4) + ((14*a
*c - (63*b*c^2)/d + (a^2*d)/b)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*d^2*(b*c - a*d
)) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b*d^2) + (5*(b*c - a*d)^2*(63*b^2*c^2 -
14*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/
(64*b^(3/2)*d^(11/2))

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Rubi in Sympy [A]  time = 59.9697, size = 289, normalized size = 0.94 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{7}{2}}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}{4 b d^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{24 b d^{3} \left (a d - b c\right )} - \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{96 b d^{4}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right )}{64 b d^{5}} - \frac{5 \left (a d - b c\right )^{2} \left (a^{2} d^{2} + 14 a b c d - 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c**2*(a + b*x)**(7/2)/(d**2*sqrt(c + d*x)*(a*d - b*c)) + (a + b*x)**(7/2)*sqr
t(c + d*x)/(4*b*d**2) - (a + b*x)**(5/2)*sqrt(c + d*x)*(a**2*d**2 + 14*a*b*c*d -
63*b**2*c**2)/(24*b*d**3*(a*d - b*c)) - 5*(a + b*x)**(3/2)*sqrt(c + d*x)*(a**2*
d**2 + 14*a*b*c*d - 63*b**2*c**2)/(96*b*d**4) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a
*d - b*c)*(a**2*d**2 + 14*a*b*c*d - 63*b**2*c**2)/(64*b*d**5) - 5*(a*d - b*c)**2
*(a**2*d**2 + 14*a*b*c*d - 63*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sq
rt(c + d*x)))/(64*b**(3/2)*d**(11/2))

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Mathematica [A]  time = 0.269729, size = 242, normalized size = 0.78 \[ \frac{5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{11/2}}+\frac{\sqrt{a+b x} \left (15 a^3 d^3 (c+d x)+a^2 b d^2 \left (-839 c^2-337 c d x+118 d^2 x^2\right )+a b^2 d \left (1785 c^3+637 c^2 d x-244 c d^2 x^2+136 d^3 x^3\right )-3 b^3 \left (315 c^4+105 c^3 d x-42 c^2 d^2 x^2+24 c d^3 x^3-16 d^4 x^4\right )\right )}{192 b d^5 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(15*a^3*d^3*(c + d*x) + a^2*b*d^2*(-839*c^2 - 337*c*d*x + 118*d^2
*x^2) + a*b^2*d*(1785*c^3 + 637*c^2*d*x - 244*c*d^2*x^2 + 136*d^3*x^3) - 3*b^3*(
315*c^4 + 105*c^3*d*x - 42*c^2*d^2*x^2 + 24*c*d^3*x^3 - 16*d^4*x^4)))/(192*b*d^5
*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(63*b^2*c^2 - 14*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]])/(128*b^(3/2)*d^
(11/2))

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Maple [B]  time = 0.04, size = 961, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(b*x+a)^(1/2)*(-96*x^4*b^3*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-272*x^
3*a*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+144*x^3*b^3*c*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^4*d^5+180*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*c*d^4-1350*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^2*d^3+2100*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^3*d^2
-945*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^4*d-236*x^2*a^2*b*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+488*x^2*a*b^2
*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-252*x^2*b^3*c^2*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^4*c*d^4+180*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^2*d^3-1350*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^3*d^2+2100*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^4*d-945*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^4*c^5-
30*x*a^3*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+674*x*a^2*b*c*d^3*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)-1274*x*a*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+
630*x*b^3*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a^3*c*d^3*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)+1678*a^2*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-3570
*a*b^2*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1890*b^3*c^4*((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/(d*x+c)^(1/2)/d^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08218, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{4} x^{4} - 945 \, b^{3} c^{4} + 1785 \, a b^{2} c^{3} d - 839 \, a^{2} b c^{2} d^{2} + 15 \, a^{3} c d^{3} - 8 \,{\left (9 \, b^{3} c d^{3} - 17 \, a b^{2} d^{4}\right )} x^{3} + 2 \,{\left (63 \, b^{3} c^{2} d^{2} - 122 \, a b^{2} c d^{3} + 59 \, a^{2} b d^{4}\right )} x^{2} -{\left (315 \, b^{3} c^{3} d - 637 \, a b^{2} c^{2} d^{2} + 337 \, a^{2} b c d^{3} - 15 \, a^{3} d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (63 \, b^{4} c^{5} - 140 \, a b^{3} c^{4} d + 90 \, a^{2} b^{2} c^{3} d^{2} - 12 \, a^{3} b c^{2} d^{3} - a^{4} c d^{4} +{\left (63 \, b^{4} c^{4} d - 140 \, a b^{3} c^{3} d^{2} + 90 \, a^{2} b^{2} c^{2} d^{3} - 12 \, a^{3} b c d^{4} - a^{4} d^{5}\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 )}{768 \,{\left (b d^{6} x + b c d^{5}\right )} \sqrt{b d}}, \frac{2 \,{\left (48 \, b^{3} d^{4} x^{4} - 945 \, b^{3} c^{4} + 1785 \, a b^{2} c^{3} d - 839 \, a^{2} b c^{2} d^{2} + 15 \, a^{3} c d^{3} - 8 \,{\left (9 \, b^{3} c d^{3} - 17 \, a b^{2} d^{4}\right )} x^{3} + 2 \,{\left (63 \, b^{3} c^{2} d^{2} - 122 \, a b^{2} c d^{3} + 59 \, a^{2} b d^{4}\right )} x^{2} -{\left (315 \, b^{3} c^{3} d - 637 \, a b^{2} c^{2} d^{2} + 337 \, a^{2} b c d^{3} - 15 \, a^{3} d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (63 \, b^{4} c^{5} - 140 \, a b^{3} c^{4} d + 90 \, a^{2} b^{2} c^{3} d^{2} - 12 \, a^{3} b c^{2} d^{3} - a^{4} c d^{4} +{\left (63 \, b^{4} c^{4} d - 140 \, a b^{3} c^{3} d^{2} + 90 \, a^{2} b^{2} c^{2} d^{3} - 12 \, a^{3} b c d^{4} - a^{4} d^{5}\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 )}{384 \,{\left (b d^{6} x + b c d^{5}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b^3*d^4*x^4 - 945*b^3*c^4 + 1785*a*b^2*c^3*d - 839*a^2*b*c^2*d^2 +
15*a^3*c*d^3 - 8*(9*b^3*c*d^3 - 17*a*b^2*d^4)*x^3 + 2*(63*b^3*c^2*d^2 - 122*a*b
^2*c*d^3 + 59*a^2*b*d^4)*x^2 - (315*b^3*c^3*d - 637*a*b^2*c^2*d^2 + 337*a^2*b*c*
d^3 - 15*a^3*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(63*b^4*c^5 - 14
0*a*b^3*c^4*d + 90*a^2*b^2*c^3*d^2 - 12*a^3*b*c^2*d^3 - a^4*c*d^4 + (63*b^4*c^4*
d - 140*a*b^3*c^3*d^2 + 90*a^2*b^2*c^2*d^3 - 12*a^3*b*c*d^4 - a^4*d^5)*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^6*x
+ b*c*d^5)*sqrt(b*d)), 1/384*(2*(48*b^3*d^4*x^4 - 945*b^3*c^4 + 1785*a*b^2*c^3*
d - 839*a^2*b*c^2*d^2 + 15*a^3*c*d^3 - 8*(9*b^3*c*d^3 - 17*a*b^2*d^4)*x^3 + 2*(6
3*b^3*c^2*d^2 - 122*a*b^2*c*d^3 + 59*a^2*b*d^4)*x^2 - (315*b^3*c^3*d - 637*a*b^2
*c^2*d^2 + 337*a^2*b*c*d^3 - 15*a^3*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x +
c) + 15*(63*b^4*c^5 - 140*a*b^3*c^4*d + 90*a^2*b^2*c^3*d^2 - 12*a^3*b*c^2*d^3 -
a^4*c*d^4 + (63*b^4*c^4*d - 140*a*b^3*c^3*d^2 + 90*a^2*b^2*c^2*d^3 - 12*a^3*b*c*
d^4 - a^4*d^5)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqr
t(d*x + c)*b*d)))/((b*d^6*x + b*c*d^5)*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)**(5/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273842, size = 568, normalized size = 1.84 \[ \frac{{\left ({\left (2 \,{\left (4 \,{\left (\frac{6 \,{\left (b x + a\right )} b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}} - \frac{9 \, b^{3} c d^{7} + 7 \, a b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} + \frac{63 \, b^{4} c^{2} d^{6} - 14 \, a b^{3} c d^{7} - a^{2} b^{2} d^{8}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} - \frac{5 \,{\left (63 \, b^{5} c^{3} d^{5} - 77 \, a b^{4} c^{2} d^{6} + 13 \, a^{2} b^{3} c d^{7} + a^{3} b^{2} d^{8}\right )}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (63 \, b^{6} c^{4} d^{4} - 140 \, a b^{5} c^{3} d^{5} + 90 \, a^{2} b^{4} c^{2} d^{6} - 12 \, a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )}}{b^{12} c d^{10} - a b^{11} d^{11}}\right )} \sqrt{b x + a}}{8257536 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{5 \,{\left (63 \, b^{3} c^{3} - 77 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{2752512 \, \sqrt{b d} b^{9} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8257536*((2*(4*(6*(b*x + a)*b^2*d^8/(b^12*c*d^10 - a*b^11*d^11) - (9*b^3*c*d^7
+ 7*a*b^2*d^8)/(b^12*c*d^10 - a*b^11*d^11))*(b*x + a) + (63*b^4*c^2*d^6 - 14*a*
b^3*c*d^7 - a^2*b^2*d^8)/(b^12*c*d^10 - a*b^11*d^11))*(b*x + a) - 5*(63*b^5*c^3*
d^5 - 77*a*b^4*c^2*d^6 + 13*a^2*b^3*c*d^7 + a^3*b^2*d^8)/(b^12*c*d^10 - a*b^11*d
^11))*(b*x + a) - 15*(63*b^6*c^4*d^4 - 140*a*b^5*c^3*d^5 + 90*a^2*b^4*c^2*d^6 -
12*a^3*b^3*c*d^7 - a^4*b^2*d^8)/(b^12*c*d^10 - a*b^11*d^11))*sqrt(b*x + a)/sqrt(
b^2*c + (b*x + a)*b*d - a*b*d) - 5/2752512*(63*b^3*c^3 - 77*a*b^2*c^2*d + 13*a^2
*b*c*d^2 + a^3*d^3)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)))/(sqrt(b*d)*b^9*d^6)