3.1641 \(\int \frac{(c+d x)^{3/4}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=270 \[ \frac{3 d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt{a+b x} (b c-a d)^{5/4}}-\frac{3 d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt{a+b x} (b c-a d)^{5/4}}+\frac{3 d^2 (c+d x)^{3/4}}{10 b \sqrt{a+b x} (b c-a d)^2}-\frac{d (c+d x)^{3/4}}{5 b (a+b x)^{3/2} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}} \]

[Out]

(-2*(c + d*x)^(3/4))/(5*b*(a + b*x)^(5/2)) - (d*(c + d*x)^(3/4))/(5*b*(b*c - a*d
)*(a + b*x)^(3/2)) + (3*d^2*(c + d*x)^(3/4))/(10*b*(b*c - a*d)^2*Sqrt[a + b*x])
- (3*d^2*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])/(10*b^(7/4)*(b*c - a*d)^(5/4)*Sqrt[a + b*x]) + (
3*d^2*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])/(10*b^(7/4)*(b*c - a*d)^(5/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.781017, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{3 d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt{a+b x} (b c-a d)^{5/4}}-\frac{3 d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt{a+b x} (b c-a d)^{5/4}}+\frac{3 d^2 (c+d x)^{3/4}}{10 b \sqrt{a+b x} (b c-a d)^2}-\frac{d (c+d x)^{3/4}}{5 b (a+b x)^{3/2} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/4)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(3/4))/(5*b*(a + b*x)^(5/2)) - (d*(c + d*x)^(3/4))/(5*b*(b*c - a*d
)*(a + b*x)^(3/2)) + (3*d^2*(c + d*x)^(3/4))/(10*b*(b*c - a*d)^2*Sqrt[a + b*x])
- (3*d^2*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])/(10*b^(7/4)*(b*c - a*d)^(5/4)*Sqrt[a + b*x]) + (
3*d^2*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])/(10*b^(7/4)*(b*c - a*d)^(5/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 102.207, size = 454, normalized size = 1.68 \[ \frac{3 d^{2} \left (c + d x\right )^{\frac{3}{4}}}{10 b \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{d \left (c + d x\right )^{\frac{3}{4}}}{5 b \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (c + d x\right )^{\frac{3}{4}}}{5 b \left (a + b x\right )^{\frac{5}{2}}} - \frac{3 d^{3} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{10 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{3 d^{2} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{7}{4}} \left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{3 d^{2} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{20 b^{\frac{7}{4}} \left (a d - b c\right )^{\frac{5}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/4)/(b*x+a)**(7/2),x)

[Out]

3*d**2*(c + d*x)**(3/4)/(10*b*sqrt(a + b*x)*(a*d - b*c)**2) + d*(c + d*x)**(3/4)
/(5*b*(a + b*x)**(3/2)*(a*d - b*c)) - 2*(c + d*x)**(3/4)/(5*b*(a + b*x)**(5/2))
- 3*d**3*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(10*b**(3/2)*(a*d - b*
c)**(5/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + 3*d**2*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))*(sq
rt(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)/(10*b**(7/4)*(a*d - b*c)**(5/4)*sqrt(a - b*c/d +
b*(c + d*x)/d)) - 3*d**2*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sq
rt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*(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)/(20*b**(
7/4)*(a*d - b*c)**(5/4)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.326375, size = 140, normalized size = 0.52 \[ \frac{(c+d x)^{3/4} \left (a^2 d^2-d^2 (a+b x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+2 a b d (3 c+4 d x)+b^2 \left (-\left (4 c^2+2 c d x-3 d^2 x^2\right )\right )\right )}{10 b (a+b x)^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/4)/(a + b*x)^(7/2),x]

[Out]

((c + d*x)^(3/4)*(a^2*d^2 + 2*a*b*d*(3*c + 4*d*x) - b^2*(4*c^2 + 2*c*d*x - 3*d^2
*x^2) - d^2*(a + b*x)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/2
, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(10*b*(b*c - a*d)^2*(a + b*x)^(5/2))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{3}{4}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/4)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(3/4)/(b*x+a)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(3/4)/(b*x + a)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x\right )^{\frac{3}{4}}}{\left (a + b x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(3/4)/(b*x+a)**(7/2),x)

[Out]

Integral((c + d*x)**(3/4)/(a + b*x)**(7/2), x)

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GIAC/XCAS [A]  time = 0.503108, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done