3.1647 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{6 b^{9/4} \sqrt{a+b x} (b c-a d)^{3/4}}-\frac{d^2 \sqrt [4]{c+d x}}{6 b^2 \sqrt{a+b x} (b c-a d)}-\frac{d \sqrt [4]{c+d x}}{3 b^2 (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/4}}{5 b (a+b x)^{5/2}} \]

[Out]

-(d*(c + d*x)^(1/4))/(3*b^2*(a + b*x)^(3/2)) - (d^2*(c + d*x)^(1/4))/(6*b^2*(b*c
- a*d)*Sqrt[a + b*x]) - (2*(c + d*x)^(5/4))/(5*b*(a + b*x)^(5/2)) - (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])/(6*b^(9/4)*(b*c - a*d)^(3/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.249855, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{d^2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{6 b^{9/4} \sqrt{a+b x} (b c-a d)^{3/4}}-\frac{d^2 \sqrt [4]{c+d x}}{6 b^2 \sqrt{a+b x} (b c-a d)}-\frac{d \sqrt [4]{c+d x}}{3 b^2 (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/4}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/4)/(a + b*x)^(7/2),x]

[Out]

-(d*(c + d*x)^(1/4))/(3*b^2*(a + b*x)^(3/2)) - (d^2*(c + d*x)^(1/4))/(6*b^2*(b*c
- a*d)*Sqrt[a + b*x]) - (2*(c + d*x)^(5/4))/(5*b*(a + b*x)^(5/2)) - (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])/(6*b^(9/4)*(b*c - a*d)^(3/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 38.2771, size = 223, normalized size = 1.27 \[ - \frac{2 \left (c + d x\right )^{\frac{5}{4}}}{5 b \left (a + b x\right )^{\frac{5}{2}}} + \frac{d^{2} \sqrt [4]{c + d x}}{6 b^{2} \sqrt{a + b x} \left (a d - b c\right )} - \frac{d \sqrt [4]{c + d x}}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}}} + \frac{d^{2} \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{12 b^{\frac{9}{4}} \left (a d - b c\right )^{\frac{3}{4}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(7/2),x)

[Out]

-2*(c + d*x)**(5/4)/(5*b*(a + b*x)**(5/2)) + d**2*(c + d*x)**(1/4)/(6*b**2*sqrt(
a + b*x)*(a*d - b*c)) - d*(c + d*x)**(1/4)/(3*b**2*(a + b*x)**(3/2)) + 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))*(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)/(12*b**(9/4)*(a*d - b*c)**(3/4)*sqrt(
a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.366704, size = 138, normalized size = 0.79 \[ \frac{\sqrt [4]{c+d x} \left (-5 a^2 d^2+5 d^2 (a+b x)^2 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )-2 a b d (c+6 d x)+b^2 \left (12 c^2+22 c d x+5 d^2 x^2\right )\right )}{30 b^2 (a+b x)^{5/2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/4)/(a + b*x)^(7/2),x]

[Out]

((c + d*x)^(1/4)*(-5*a^2*d^2 - 2*a*b*d*(c + 6*d*x) + b^2*(12*c^2 + 22*c*d*x + 5*
d^2*x^2) + 5*d^2*(a + b*x)^2*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F
1[1/4, 1/2, 5/4, (b*(c + d*x))/(b*c - a*d)]))/(30*b^2*(-(b*c) + a*d)*(a + b*x)^(
5/2))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(5/4)/(b*x+a)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(7/2),x, algorithm="fricas")

[Out]

integral((d*x + c)^(5/4)/((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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/4)/(b*x+a)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.463258, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done