3.1679 \(\int (a+b x)^{3/4} (c+d x)^{5/4} \, dx\)

Optimal. Leaf size=205 \[ \frac{5 (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac{5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \]

[Out]

(5*(b*c - a*d)^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(96*b^2*d) + (5*(b*c - a*d)*(a
+ b*x)^(7/4)*(c + d*x)^(1/4))/(24*b^2) + ((a + b*x)^(7/4)*(c + d*x)^(5/4))/(3*b
) + (5*(b*c - a*d)^3*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))]
)/(64*b^(9/4)*d^(7/4)) - (5*(b*c - a*d)^3*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(
1/4)*(c + d*x)^(1/4))])/(64*b^(9/4)*d^(7/4))

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Rubi [A]  time = 0.269895, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{5 (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac{5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]

[Out]

(5*(b*c - a*d)^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(96*b^2*d) + (5*(b*c - a*d)*(a
+ b*x)^(7/4)*(c + d*x)^(1/4))/(24*b^2) + ((a + b*x)^(7/4)*(c + d*x)^(5/4))/(3*b
) + (5*(b*c - a*d)^3*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))]
)/(64*b^(9/4)*d^(7/4)) - (5*(b*c - a*d)^3*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(
1/4)*(c + d*x)^(1/4))])/(64*b^(9/4)*d^(7/4))

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Rubi in Sympy [A]  time = 37.123, size = 184, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{9}{4}}}{3 d} + \frac{\left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{5}{4}} \left (a d - b c\right )}{8 b d} - \frac{5 \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x} \left (a d - b c\right )^{2}}{32 b^{2} d} + \frac{5 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{64 b^{\frac{9}{4}} d^{\frac{7}{4}}} + \frac{5 \left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{64 b^{\frac{9}{4}} d^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/4)*(d*x+c)**(5/4),x)

[Out]

(a + b*x)**(3/4)*(c + d*x)**(9/4)/(3*d) + (a + b*x)**(3/4)*(c + d*x)**(5/4)*(a*d
- b*c)/(8*b*d) - 5*(a + b*x)**(3/4)*(c + d*x)**(1/4)*(a*d - b*c)**2/(32*b**2*d)
+ 5*(a*d - b*c)**3*atan(b**(1/4)*(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/
(64*b**(9/4)*d**(7/4)) + 5*(a*d - b*c)**3*atanh(b**(1/4)*(c + d*x)**(1/4)/(d**(1
/4)*(a + b*x)**(1/4)))/(64*b**(9/4)*d**(7/4))

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Mathematica [C]  time = 0.275688, size = 143, normalized size = 0.7 \[ \frac{\sqrt [4]{c+d x} \left (-d (a+b x) \left (15 a^2 d^2-6 a b d (7 c+2 d x)+b^2 \left (-\left (5 c^2+52 c d x+32 d^2 x^2\right )\right )\right )-15 (b c-a d)^3 \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{96 b^2 d^2 \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]

[Out]

((c + d*x)^(1/4)*(-(d*(a + b*x)*(15*a^2*d^2 - 6*a*b*d*(7*c + 2*d*x) - b^2*(5*c^2
+ 52*c*d*x + 32*d^2*x^2))) - 15*(b*c - a*d)^3*((d*(a + b*x))/(-(b*c) + a*d))^(1
/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(96*b^2*d^2*(a
+ b*x)^(1/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{3}{4}}} \left ( dx+c \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/4)*(d*x+c)^(5/4),x)

[Out]

int((b*x+a)^(3/4)*(d*x+c)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/4)*(d*x + c)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/4)*(d*x + c)^(5/4), x)

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Fricas [A]  time = 0.261882, size = 2225, normalized size = 10.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/4)*(d*x + c)^(5/4),x, algorithm="fricas")

[Out]

-1/384*(60*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3
*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 -
792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c
^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4)*arctan(-(b^3*d^2*x + a*
b^2*d^2)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9
*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7
*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10
- 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a
^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b*x + a)*sqrt(((b^6*c^6
- 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4
- 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (b^5*d^4*x + a*b^4*d^4)
*sqrt((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3
+ 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5
*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12
*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7)))/(b*x + a)))) + 15*b^2*d*((b^12*c^12 - 12
*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^
4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^
4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12
*d^12)/(b^9*d^7))^(1/4)*log(-5*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b^3*d^2*x + a*b^2*d^2)*((b^12*c^12 - 12*a
*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4
- 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*
c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d
^12)/(b^9*d^7))^(1/4))/(b*x + a)) - 15*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66
*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7
*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9
*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(
1/4)*log(-5*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/4)
*(d*x + c)^(1/4) - (b^3*d^2*x + a*b^2*d^2)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a
^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d
^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b
^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/
4))/(b*x + a)) - 4*(32*b^2*d^2*x^2 + 5*b^2*c^2 + 42*a*b*c*d - 15*a^2*d^2 + 4*(13
*b^2*c*d + 3*a*b*d^2)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4))/(b^2*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((b*x+a)**(3/4)*(d*x+c)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/4)*(d*x + c)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/4)*(d*x + c)^(5/4), x)