3.781 \(\int \frac{x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=497 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{240 b^5 d^2 (b c-a d)}+\frac{(b c-a d) \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{15/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^7 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right )}{192 b^6 d^2 (b c-a d)}+\frac{x^2 \sqrt{a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{15 b^3 (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

[Out]

((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4
*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^7*d^2) + ((3*b^4*c^4 + 28*a*b^3*c^3*d
+ 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqrt[a + b*x]*(c + d*x)
^(3/2))/(192*b^6*d^2*(b*c - a*d)) - (2*x^4*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)
) - (2*(8*b*c - 13*a*d)*x^3*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) +
((93*b*c - 143*a*d)*x^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(15*b^3*(b*c - a*d)) - (
Sqrt[a + b*x]*(c + d*x)^(5/2)*(15*b^3*c^3 + 125*a*b^2*c^2*d - 2343*a^2*b*c*d^2 +
3003*a^3*d^3 - 2*b*d*(15*b^2*c^2 - 902*a*b*c*d + 1287*a^2*d^2)*x))/(240*b^5*d^2
*(b*c - a*d)) + ((b*c - a*d)*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 -
2772*a^3*b*c*d^3 + 3003*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[
c + d*x])])/(128*b^(15/2)*d^(5/2))

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Rubi [A]  time = 1.346, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{240 b^5 d^2 (b c-a d)}+\frac{(b c-a d) \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{15/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^7 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right )}{192 b^6 d^2 (b c-a d)}+\frac{x^2 \sqrt{a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{15 b^3 (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4
*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^7*d^2) + ((3*b^4*c^4 + 28*a*b^3*c^3*d
+ 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqrt[a + b*x]*(c + d*x)
^(3/2))/(192*b^6*d^2*(b*c - a*d)) - (2*x^4*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)
) - (2*(8*b*c - 13*a*d)*x^3*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) +
((93*b*c - 143*a*d)*x^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(15*b^3*(b*c - a*d)) - (
Sqrt[a + b*x]*(c + d*x)^(5/2)*(15*b^3*c^3 + 125*a*b^2*c^2*d - 2343*a^2*b*c*d^2 +
3003*a^3*d^3 - 2*b*d*(15*b^2*c^2 - 902*a*b*c*d + 1287*a^2*d^2)*x))/(240*b^5*d^2
*(b*c - a*d)) + ((b*c - a*d)*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 -
2772*a^3*b*c*d^3 + 3003*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[
c + d*x])])/(128*b^(15/2)*d^(5/2))

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Rubi in Sympy [A]  time = 127.19, size = 430, normalized size = 0.87 \[ - \frac{2 x^{4} \left (c + d x\right )^{\frac{5}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{11 x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (39 a d - 23 b c\right )}{40 b^{4}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{45045 a^{3} d^{3}}{32} - \frac{32571 a^{2} b c d^{2}}{32} + \frac{1215 a b^{2} c^{2} d}{32} + \frac{135 b^{3} c^{3}}{32} - \frac{9 b d x \left (1001 a^{2} d^{2} - 638 a b c d + 5 b^{2} c^{2}\right )}{8}\right )}{90 b^{6} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3003 a^{4} d^{4} - 2772 a^{3} b c d^{3} + 378 a^{2} b^{2} c^{2} d^{2} + 28 a b^{3} c^{3} d + 3 b^{4} c^{4}\right )}{128 b^{7} d^{2}} - \frac{\left (a d - b c\right ) \left (3003 a^{4} d^{4} - 2772 a^{3} b c d^{3} + 378 a^{2} b^{2} c^{2} d^{2} + 28 a b^{3} c^{3} d + 3 b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{15}{2}} d^{\frac{5}{2}}} - \frac{2 x^{4} \left (c + d x\right )^{\frac{3}{2}} \left (13 a d - 8 b c\right )}{3 a b^{2} \sqrt{a + b x}} + \frac{x^{3} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (143 a d - 80 b c\right )}{15 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(d*x+c)**(5/2)/(b*x+a)**(5/2),x)

[Out]

-2*x**4*(c + d*x)**(5/2)/(3*b*(a + b*x)**(3/2)) - 11*x**2*sqrt(a + b*x)*(c + d*x
)**(3/2)*(39*a*d - 23*b*c)/(40*b**4) - sqrt(a + b*x)*(c + d*x)**(3/2)*(45045*a**
3*d**3/32 - 32571*a**2*b*c*d**2/32 + 1215*a*b**2*c**2*d/32 + 135*b**3*c**3/32 -
9*b*d*x*(1001*a**2*d**2 - 638*a*b*c*d + 5*b**2*c**2)/8)/(90*b**6*d**2) + sqrt(a
+ b*x)*sqrt(c + d*x)*(3003*a**4*d**4 - 2772*a**3*b*c*d**3 + 378*a**2*b**2*c**2*d
**2 + 28*a*b**3*c**3*d + 3*b**4*c**4)/(128*b**7*d**2) - (a*d - b*c)*(3003*a**4*d
**4 - 2772*a**3*b*c*d**3 + 378*a**2*b**2*c**2*d**2 + 28*a*b**3*c**3*d + 3*b**4*c
**4)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(128*b**(15/2)*d**(5/2
)) - 2*x**4*(c + d*x)**(3/2)*(13*a*d - 8*b*c)/(3*a*b**2*sqrt(a + b*x)) + x**3*sq
rt(a + b*x)*(c + d*x)**(3/2)*(143*a*d - 80*b*c)/(15*a*b**3)

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Mathematica [A]  time = 0.582638, size = 385, normalized size = 0.77 \[ \frac{(b c-a d) \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{15/2} d^{5/2}}+\frac{\sqrt{c+d x} \left (45045 a^6 d^4+2310 a^5 b d^3 (26 d x-31 c)+21 a^4 b^2 d^2 \left (1304 c^2-4642 c d x+429 d^2 x^2\right )-6 a^3 b^3 d \left (65 c^3-6441 c^2 d x+2673 c d^2 x^2+429 d^3 x^3\right )+a^2 b^4 \left (-45 c^4-750 c^3 d x+7404 c^2 d^2 x^2+4378 c d^3 x^3+1144 d^4 x^4\right )-2 a b^5 x \left (45 c^4+165 c^3 d x+917 c^2 d^2 x^2+944 c d^3 x^3+312 d^4 x^4\right )+3 b^6 x^2 \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )\right )}{1920 b^7 d^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[c + d*x]*(45045*a^6*d^4 + 2310*a^5*b*d^3*(-31*c + 26*d*x) + 21*a^4*b^2*d^2
*(1304*c^2 - 4642*c*d*x + 429*d^2*x^2) - 6*a^3*b^3*d*(65*c^3 - 6441*c^2*d*x + 26
73*c*d^2*x^2 + 429*d^3*x^3) + 3*b^6*x^2*(-15*c^4 + 10*c^3*d*x + 248*c^2*d^2*x^2
+ 336*c*d^3*x^3 + 128*d^4*x^4) - 2*a*b^5*x*(45*c^4 + 165*c^3*d*x + 917*c^2*d^2*x
^2 + 944*c*d^3*x^3 + 312*d^4*x^4) + a^2*b^4*(-45*c^4 - 750*c^3*d*x + 7404*c^2*d^
2*x^2 + 4378*c*d^3*x^3 + 1144*d^4*x^4)))/(1920*b^7*d^2*(a + b*x)^(3/2)) + ((b*c
- a*d)*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 30
03*a^4*d^4)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d
*x]])/(256*b^(15/2)*d^(5/2))

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Maple [B]  time = 0.059, size = 1762, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x+c)^(5/2)/(b*x+a)^(5/2),x)

[Out]

-1/3840*(d*x+c)^(1/2)*(90*a^2*b^4*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+45045*
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^5*b^2*d^5+90090*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^6*b*d^5-90*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*b^6*c^5-86625*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^6*b*c*d^4+47250*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^5*b^2*c^2*d^3-5250*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*b^3*c^
3*d^2-375*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^4*c^4*d-768*x^6*b^6*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-10500*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^4*c^3*d^2-90090*a^6*d^4*((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))*x^2*b^7*c^5-45*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^5*
c^5-8756*x^3*a^2*b^4*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3668*x^3*a*b^5*c^
2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+32076*x^2*a^3*b^3*c*d^3*((b*x+a)*(d*x+
c))^(1/2)*(b*d)^(1/2)-14808*x^2*a^2*b^4*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+660*x^2*a*b^5*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+194964*x*a^4*b^2*c*d
^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-77292*x*a^3*b^3*c^2*d^2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+1500*x*a^2*b^4*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3776
*x^4*a*b^5*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+90*x^2*b^6*c^4*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)-120120*x*a^5*b*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1
80*x*a*b^5*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+143220*a^5*b*c*d^3*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)-2288*x^4*a^2*b^4*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)-1488*x^4*b^6*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+5148*x^3*a^3*b^3*d^4
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-60*x^3*b^6*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)-18018*x^2*a^4*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-750*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^5*c^
4*d-54768*a^4*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+780*a^3*b^3*c^3*d*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-86625*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^4*b^3*c*d^4+47250*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^3*b^4*c^2*d^3-52
50*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^5*c^3*d^2-375*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^6*c^4*d-173250*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^5*b^2*c*d^4+94500*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*b^3*c^2*d^3+1248*x
^5*a*b^5*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2016*x^5*b^6*c*d^3*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+45045*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^7*d^5)/((b*x+a)*(d*x+c))^(1/2)/d^2/(b*d)^(1/2)/(b*x+
a)^(3/2)/b^7

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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^4/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.4814, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)*x^4/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*b^6*d^4*x^6 - 45*a^2*b^4*c^4 - 390*a^3*b^3*c^3*d + 27384*a^4*b^2
*c^2*d^2 - 71610*a^5*b*c*d^3 + 45045*a^6*d^4 + 48*(21*b^6*c*d^3 - 13*a*b^5*d^4)*
x^5 + 8*(93*b^6*c^2*d^2 - 236*a*b^5*c*d^3 + 143*a^2*b^4*d^4)*x^4 + 2*(15*b^6*c^3
*d - 917*a*b^5*c^2*d^2 + 2189*a^2*b^4*c*d^3 - 1287*a^3*b^3*d^4)*x^3 - 3*(15*b^6*
c^4 + 110*a*b^5*c^3*d - 2468*a^2*b^4*c^2*d^2 + 5346*a^3*b^3*c*d^3 - 3003*a^4*b^2
*d^4)*x^2 - 6*(15*a*b^5*c^4 + 125*a^2*b^4*c^3*d - 6441*a^3*b^3*c^2*d^2 + 16247*a
^4*b^2*c*d^3 - 10010*a^5*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(3
*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c^3*d^2 - 3150*a^5*b^2*c^2*d^3 + 5
775*a^6*b*c*d^4 - 3003*a^7*d^5 + (3*b^7*c^5 + 25*a*b^6*c^4*d + 350*a^2*b^5*c^3*d
^2 - 3150*a^3*b^4*c^2*d^3 + 5775*a^4*b^3*c*d^4 - 3003*a^5*b^2*d^5)*x^2 + 2*(3*a*
b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4*c^3*d^2 - 3150*a^4*b^3*c^2*d^3 + 5775*a
^5*b^2*c*d^4 - 3003*a^6*b*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^9*d^2*x^2 + 2*a*b^8*d^2*x + a^2*b^7*d^2)*sqrt
(b*d)), 1/3840*(2*(384*b^6*d^4*x^6 - 45*a^2*b^4*c^4 - 390*a^3*b^3*c^3*d + 27384*
a^4*b^2*c^2*d^2 - 71610*a^5*b*c*d^3 + 45045*a^6*d^4 + 48*(21*b^6*c*d^3 - 13*a*b^
5*d^4)*x^5 + 8*(93*b^6*c^2*d^2 - 236*a*b^5*c*d^3 + 143*a^2*b^4*d^4)*x^4 + 2*(15*
b^6*c^3*d - 917*a*b^5*c^2*d^2 + 2189*a^2*b^4*c*d^3 - 1287*a^3*b^3*d^4)*x^3 - 3*(
15*b^6*c^4 + 110*a*b^5*c^3*d - 2468*a^2*b^4*c^2*d^2 + 5346*a^3*b^3*c*d^3 - 3003*
a^4*b^2*d^4)*x^2 - 6*(15*a*b^5*c^4 + 125*a^2*b^4*c^3*d - 6441*a^3*b^3*c^2*d^2 +
16247*a^4*b^2*c*d^3 - 10010*a^5*b*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)
+ 15*(3*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c^3*d^2 - 3150*a^5*b^2*c^2
*d^3 + 5775*a^6*b*c*d^4 - 3003*a^7*d^5 + (3*b^7*c^5 + 25*a*b^6*c^4*d + 350*a^2*b
^5*c^3*d^2 - 3150*a^3*b^4*c^2*d^3 + 5775*a^4*b^3*c*d^4 - 3003*a^5*b^2*d^5)*x^2 +
2*(3*a*b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4*c^3*d^2 - 3150*a^4*b^3*c^2*d^3
+ 5775*a^5*b^2*c*d^4 - 3003*a^6*b*d^5)*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^9*d^2*x^2 + 2*a*b^8*d^2*x + a^2*b^
7*d^2)*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**4*(d*x+c)**(5/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.748464, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)*x^4/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

sage0*x