3.1636 \(\int (a+b x)^{3/2} (c+d x)^{3/4} \, dx\)

Optimal. Leaf size=270 \[ -\frac{16 (b c-a d)^{15/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{65 b^{7/4} d^3 \sqrt{a+b x}}+\frac{16 (b c-a d)^{15/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{65 b^{7/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}{65 b d^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}{39 b d}+\frac{4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b} \]

[Out]

(-8*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/4))/(65*b*d^2) + (4*(b*c - a*d)*(a
+ b*x)^(3/2)*(c + d*x)^(3/4))/(39*b*d) + (4*(a + b*x)^(5/2)*(c + d*x)^(3/4))/(13
*b) + (16*(b*c - a*d)^(15/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin
[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(65*b^(7/4)*d^3*Sqrt[a + b*x
]) - (16*(b*c - a*d)^(15/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[
(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(65*b^(7/4)*d^3*Sqrt[a + b*x]
)

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Rubi [A]  time = 0.939521, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{16 (b c-a d)^{15/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{65 b^{7/4} d^3 \sqrt{a+b x}}+\frac{16 (b c-a d)^{15/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{65 b^{7/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}{65 b d^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)}{39 b d}+\frac{4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)*(c + d*x)^(3/4),x]

[Out]

(-8*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/4))/(65*b*d^2) + (4*(b*c - a*d)*(a
+ b*x)^(3/2)*(c + d*x)^(3/4))/(39*b*d) + (4*(a + b*x)^(5/2)*(c + d*x)^(3/4))/(13
*b) + (16*(b*c - a*d)^(15/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin
[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(65*b^(7/4)*d^3*Sqrt[a + b*x
]) - (16*(b*c - a*d)^(15/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[
(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(65*b^(7/4)*d^3*Sqrt[a + b*x]
)

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Rubi in Sympy [A]  time = 103.324, size = 456, normalized size = 1.69 \[ \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{4}}}{13 d} + \frac{8 \sqrt{a + b x} \left (c + d x\right )^{\frac{7}{4}} \left (a d - b c\right )}{39 d^{2}} + \frac{16 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}}{195 b d^{2}} - \frac{16 \sqrt [4]{c + d x} \left (a d - b c\right )^{\frac{5}{2}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{65 b^{\frac{3}{2}} d^{2} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{16 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{15}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{8 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{15}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} d^{3} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(3/4),x)

[Out]

4*(a + b*x)**(3/2)*(c + d*x)**(7/4)/(13*d) + 8*sqrt(a + b*x)*(c + d*x)**(7/4)*(a
*d - b*c)/(39*d**2) + 16*sqrt(a + b*x)*(c + d*x)**(3/4)*(a*d - b*c)**2/(195*b*d*
*2) - 16*(c + d*x)**(1/4)*(a*d - b*c)**(5/2)*sqrt(a - b*c/d + b*(c + d*x)/d)/(65
*b**(3/2)*d**2*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 16*sqrt((a*d - b*c
+ b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(a
*d - b*c)**(15/4)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_e(2*atan(
b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(65*b**(7/4)*d**3*sqrt(a - b
*c/d + b*(c + d*x)/d)) - 8*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*
sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(a*d - b*c)**(15/4)*(sqrt(b)*sqrt(c + d*
x)/sqrt(a*d - b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)*
*(1/4)), 1/2)/(65*b**(7/4)*d**3*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.271283, size = 141, normalized size = 0.52 \[ -\frac{4 (c+d x)^{3/4} \left (-d (a+b x) \left (4 a^2 d^2+a b d (17 c+25 d x)+b^2 \left (-6 c^2+5 c d x+15 d^2 x^2\right )\right )-4 (b c-a d)^3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{195 b d^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)*(c + d*x)^(3/4),x]

[Out]

(-4*(c + d*x)^(3/4)*(-(d*(a + b*x)*(4*a^2*d^2 + a*b*d*(17*c + 25*d*x) + b^2*(-6*
c^2 + 5*c*d*x + 15*d^2*x^2))) - 4*(b*c - a*d)^3*Sqrt[(d*(a + b*x))/(-(b*c) + a*d
)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(195*b*d^3*Sqrt
[a + b*x])

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(3/4),x)

[Out]

int((b*x+a)^(3/2)*(d*x+c)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*(d*x + c)^(3/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/2)*(d*x + c)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*(d*x + c)^(3/4),x, algorithm="fricas")

[Out]

integral((b*x + a)^(3/2)*(d*x + c)^(3/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(d*x+c)**(3/4),x)

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*(d*x + c)^(3/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/2)*(d*x + c)^(3/4), x)